# OEIS Recent Additions (http://oeis.org/recent.txt)
# Last Modified: February 11 08:17 UTC 2012
# Use of this content is governed by the
# OEIS End-User License: http://oeis.org/LICENSE
%I A206640
%S A206640 3,5,11,61,67,73,79
%N A206640 Primes in A060432.
%K A206640 nonn,full,fini,new
%O A206640 1,1
%A A206640 Max Alekseyev (maxale(AT)gmail.com), Feb 11 2012

%I A206625
%S A206625 0,1,1,1,2,5,5,13,16,37,45,109,130,313,377,905,1088,2617,3145,7561,
%T A206625 9090,21853,26269,63157,75920,182525,219413,527509,634114,1524529,
%U A206625 1832625,4405969,5296384,12733489,15306833,36800465,44237570,106355317
%N A206625 Expansion of x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4  - 2*x^6 + x^8) in powers of x.
%C A206625 This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m).
%D A206625 J. A. Sjogren, Cycles and spanning trees. Math. Comput. Modelling 15, No.9, 87-102 (1991).
%H A206625 J. A. Sjogren, <a href="http://jon-arny.com/httpdocs/PDF/Cycles.pdf">Cycles and Spanning Trees</a>, see equation (3.5)
%F A206625 G.f.: x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8).
%F A206625 a(-n) = a(n) = 2*a(n-2) + 2*a(n-4) + 2*a(n-6) - a(n-8).
%F A206625 a(2*n + 5) = A071100(n).  a(2*n + 6) = A071101(n). a(n + 3) = A112835(n). a(2*n) = A138573(n).
%e A206625 x + x^2 + x^3 + 2*x^4 + 5*x^5 + 5*x^6 + 13*x^7 + 16*x^8 + 37*x^9 + ...
%o A206625 (PARI) {a(n) = local(m = abs(n)); polcoeff( x * (1 + x) * (1 - x^2) * (1 + x^3) / (1 - 2*x^2 - 2*x^4 - 2*x^6 + x^8) + x * O(x^m), m)}
%o A206625 (PARI) {a(n) = local(m = abs(n), v); v = polroots( Pol([ 1, 2, 4, 2,
%o A206625 1])); sqrtint( round( prod( k=1, 4, v[k]^m - 1, 2^(m%2) / 20)))}
%Y A206625 Cf. A071100, A071101, A112835, A138573.
%K A206625 nonn,new
%O A206625 0,5
%A A206625 Michael Somos, Feb 10 2012

%I A206635
%S A206635 14,121,121,1085,3232,1085,9729,86392,86392,9729,87238,2309200,
%T A206635 6885875,2309200,87238,782246,61723264,548812835,548812835,61723264,
%U A206635 782246,7014246,1649819344,43740991174,130424683977,43740991174,1649819344
%N A206635 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206635 Table starts
%C A206635 .......14...........121..............1085..................9729
%C A206635 ......121..........3232.............86392...............2309200
%C A206635 .....1085.........86392...........6885875.............548812835
%C A206635 .....9729.......2309200.........548812835..........130424683977
%C A206635 ....87238......61723264.......43740991174........30995204476194
%C A206635 ...782246....1649819344.....3486207195767......7365962082764193
%C A206635 ..7014246...44098506880...277854703424333...1750509332449744133
%C A206635 .62895364.1178721971776.22145337882995800.416005797001053603530
%H A206635 R. H. Hardin, <a href="/A206635/b206635.txt">Table of n, a(n) for n = 1..179</a>
%e A206635 Some solutions for n=4 k=3
%e A206635 ..0..0..0..0....0..0..1..1....0..1..0..1....0..1..0..0....0..1..0..2
%e A206635 ..1..0..0..1....1..1..1..0....2..2..1..0....1..1..1..0....1..2..2..1
%e A206635 ..2..1..0..0....1..2..1..2....1..1..0..0....2..1..2..2....1..0..1..1
%e A206635 ..2..0..0..0....0..1..2..1....2..0..0..0....2..0..2..1....0..0..2..0
%e A206635 ..2..0..0..1....2..0..0..1....0..1..1..0....1..1..1..0....0..2..0..0
%K A206635 nonn,tabl,new
%O A206635 1,1
%A A206635 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206634
%S A206634 7014246,44098506880,277854703424333,1750509332449744133,
%T A206634 11028339625449228398660,69479426940659354584343140,
%U A206634 437725944874306422735306779417,2757708457712094967781824783252256
%N A206634 Number of (n+1)X8 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206634 Column 7 of A206635
%H A206634 R. H. Hardin, <a href="/A206634/b206634.txt">Table of n, a(n) for n = 1..79</a>
%e A206634 Some solutions for n=4
%e A206634 ..0..0..0..0..1..0..0..0....0..0..1..2..2..2..1..0....0..1..2..2..0..0..0..2
%e A206634 ..2..1..0..0..2..0..0..1....0..1..2..1..0..1..0..0....0..2..2..0..1..0..0..0
%e A206634 ..2..0..0..1..1..1..0..1....2..2..0..0..0..1..0..2....2..2..1..2..0..0..2..0
%e A206634 ..2..2..1..0..1..1..0..0....1..1..1..2..0..2..2..0....0..2..2..2..1..1..0..2
%e A206634 ..2..0..0..0..0..1..1..0....1..1..0..1..1..1..1..1....2..0..1..0..1..2..0..2
%K A206634 nonn,new
%O A206634 1,1
%A A206634 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206633
%S A206633 782246,1649819344,3486207195767,7365962082764193,
%T A206633 15563412880264965921,32883690237686565806902,
%U A206633 69479426940659354584343140,146801978229825187395072986335,310175569456316609985253179357234
%N A206633 Number of (n+1)X7 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206633 Column 6 of A206635
%H A206633 R. H. Hardin, <a href="/A206633/b206633.txt">Table of n, a(n) for n = 1..193</a>
%e A206633 Some solutions for n=4
%e A206633 ..0..0..0..1..2..1..2....0..0..0..0..1..0..0....0..0..1..0..0..0..2
%e A206633 ..0..0..1..0..2..0..1....2..0..0..1..2..0..0....2..1..2..0..0..2..0
%e A206633 ..2..0..1..0..0..2..1....1..1..1..0..1..2..2....0..2..2..2..1..1..0
%e A206633 ..1..2..0..0..1..2..2....1..0..1..1..0..0..2....2..2..0..1..0..1..2
%e A206633 ..2..1..2..1..0..2..2....0..0..0..0..1..1..0....0..2..1..0..2..0..1
%K A206633 nonn,new
%O A206633 1,1
%A A206633 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206632
%S A206632 87238,61723264,43740991174,30995204476194,21963397411445576,
%T A206632 15563412880264965921,11028339625449228398660,
%U A206632 7814756058919506797780720,5537589006760757757340511058,3923973029438323839541440812536
%N A206632 Number of (n+1)X6 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206632 Column 5 of A206635
%H A206632 R. H. Hardin, <a href="/A206632/b206632.txt">Table of n, a(n) for n = 1..210</a>
%e A206632 Some solutions for n=4
%e A206632 ..0..0..1..2..2..2....0..0..0..0..0..1....0..0..1..1..0..1....0..0..0..0..0..1
%e A206632 ..1..0..0..1..2..1....2..1..2..1..2..1....1..1..1..1..0..0....0..2..2..1..1..2
%e A206632 ..0..1..0..0..2..2....0..2..1..1..1..2....0..1..2..1..0..2....2..1..2..0..1..1
%e A206632 ..0..0..0..1..0..2....1..1..2..1..1..1....1..0..2..2..1..0....2..1..1..0..1..2
%e A206632 ..1..1..1..2..0..2....0..1..2..2..1..2....2..2..2..1..2..0....0..1..1..2..0..2
%K A206632 nonn,new
%O A206632 1,1
%A A206632 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206631
%S A206631 9729,2309200,548812835,130424683977,30995204476194,7365962082764193,
%T A206631 1750509332449744133,416005797001053603530,98863125127708691660391,
%U A206631 23494666613322778190102135,5583470667743721282474810860
%N A206631 Number of (n+1)X5 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206631 Column 4 of A206635
%H A206631 R. H. Hardin, <a href="/A206631/b206631.txt">Table of n, a(n) for n = 1..210</a>
%F A206631 Empirical: a(n) = 200*a(n-1) +8019*a(n-2) +212850*a(n-3) +1791825*a(n-4) +10193932*a(n-5) -153523984*a(n-6) +1400513402*a(n-7) -14595701184*a(n-8) +63361541608*a(n-9) -318507890248*a(n-10) +1072398618504*a(n-11) -2729587607696*a(n-12) +9386032798640*a(n-13) -19591166905600*a(n-14) +30705316657664*a(n-15) -71422982097664*a(n-16) +82765088625152*a(n-17) -56333612907520*a(n-18) +125995858063360*a(n-19) -11670533963776*a(n-20) -83402072440832*a(n-21) -88610306424832*a(n-22) -11808913489920*a(n-23) -14131206291456*a(n-24) -21679552069632*a(n-25) -14978761359360*a(n-26) -2076969009152*a(n-27) -270582939648*a(n-28) -401042571264*a(n-29) -173946175488*a(n-30)
%e A206631 Some solutions for n=4
%e A206631 ..0..1..2..1..0....0..0..0..0..0....0..1..0..2..1....0..0..0..0..1
%e A206631 ..1..2..1..2..1....1..2..1..0..0....2..2..1..1..0....1..2..0..1..2
%e A206631 ..1..1..1..0..2....2..0..0..2..1....1..1..0..0..2....1..2..2..0..0
%e A206631 ..2..1..2..1..2....2..1..1..1..1....0..0..2..0..0....0..2..2..0..0
%e A206631 ..2..1..2..1..1....2..0..1..2..2....0..1..2..0..1....1..0..1..1..2
%K A206631 nonn,new
%O A206631 1,1
%A A206631 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206630
%S A206630 1085,86392,6885875,548812835,43740991174,3486207195767,
%T A206630 277854703424333,22145337882995800,1765008776141025599,
%U A206630 140673219629487281855,11211816614428114906158,893594616847882973762891
%N A206630 Number of (n+1)X4 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206630 Column 3 of A206635
%H A206630 R. H. Hardin, <a href="/A206630/b206630.txt">Table of n, a(n) for n = 1..210</a>
%F A206630 Empirical: a(n) = 66*a(n-1) +970*a(n-2) +9295*a(n-3) +33266*a(n-4) +72056*a(n-5) -224448*a(n-6) +362688*a(n-7) -1135552*a(n-8) +493824*a(n-9) -311808*a(n-10) +294912*a(n-11)
%e A206630 Some solutions for n=4
%e A206630 ..0..0..0..1....0..1..2..0....0..0..0..0....0..0..0..0....0..1..0..2
%e A206630 ..2..0..1..0....1..2..2..0....0..1..2..1....0..1..0..1....2..2..0..1
%e A206630 ..0..2..1..1....0..1..1..2....0..0..2..0....1..2..2..2....2..0..1..1
%e A206630 ..2..0..0..1....0..1..2..1....0..2..1..2....1..2..2..0....2..1..0..1
%e A206630 ..2..2..2..1....2..2..0..0....1..1..1..1....1..2..1..1....1..2..2..2
%K A206630 nonn,new
%O A206630 1,1
%A A206630 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206629
%S A206629 121,3232,86392,2309200,61723264,1649819344,44098506880,1178721971776,
%T A206629 31506406571584,842144015978368,22509915310006528,601674152698239232,
%U A206629 16082325545854547968,429869213764553543680,11490100757858808902656
%N A206629 Number of (n+1)X3 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206629 Column 2 of A206635
%H A206629 R. H. Hardin, <a href="/A206629/b206629.txt">Table of n, a(n) for n = 1..210</a>
%F A206629 Empirical: a(n) = 24*a(n-1) +68*a(n-2) +132*a(n-3) +8*a(n-4) +48*a(n-5)
%e A206629 Some solutions for n=4
%e A206629 ..0..0..0....0..0..1....0..0..0....0..0..0....0..0..1....0..0..1....0..1..0
%e A206629 ..0..1..1....0..0..0....0..1..0....0..1..0....2..0..2....1..1..1....2..1..2
%e A206629 ..1..0..1....2..2..1....2..0..1....0..0..2....1..0..2....1..0..1....2..1..1
%e A206629 ..2..2..1....2..0..0....2..0..1....1..0..0....1..2..1....2..1..2....0..1..1
%e A206629 ..0..1..1....2..0..0....0..0..2....2..0..0....2..2..1....0..1..2....0..0..2
%K A206629 nonn,new
%O A206629 1,1
%A A206629 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206628
%S A206628 14,121,1085,9729,87238,782246,7014246,62895364,563970356,5057011236,
%T A206628 45345224920,406601711384,3645917557848,32692225503376,
%U A206628 293144754758096,2628571347438864,23569882170595936,211346496672826976
%N A206628 Number of (n+1)X2 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206628 Column 1 of A206635
%H A206628 R. H. Hardin, <a href="/A206628/b206628.txt">Table of n, a(n) for n = 1..210</a>
%F A206628 Empirical: a(n) = 8*a(n-1) +8*a(n-2) +6*a(n-3) for n>4
%e A206628 Some solutions for n=4
%e A206628 ..0..1....0..1....0..0....0..1....0..1....0..0....0..0....0..1....0..1....0..1
%e A206628 ..0..2....0..0....1..1....2..0....2..0....0..0....0..1....1..1....0..2....0..2
%e A206628 ..2..0....1..1....1..1....1..2....0..1....0..1....2..1....0..1....1..2....1..1
%e A206628 ..0..2....1..0....1..0....2..0....2..2....0..1....2..0....2..2....2..1....1..2
%e A206628 ..1..0....2..1....1..2....0..1....2..1....1..2....0..1....1..2....1..0....1..1
%K A206628 nonn,new
%O A206628 1,1
%A A206628 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206627
%S A206627 14,3232,6885875,130424683977,21963397411445576,
%T A206627 32883690237686565806902,437725944874306422735306779417,
%U A206627 51804179019799430182581262620045135320,54508981434487150136168842178572782994896876244
%N A206627 Number of (n+1)X(n+1) 0..2 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..2 introduced in row major order
%C A206627 Diagonal of A206635
%e A206627 Some solutions for n=4
%e A206627 ..0..0..0..0..0....0..0..1..2..2....0..0..0..1..2....0..0..0..0..0
%e A206627 ..0..1..0..2..1....2..2..1..1..1....2..1..0..0..1....1..2..1..0..0
%e A206627 ..1..1..2..0..2....1..0..0..0..0....1..1..0..2..1....2..0..0..2..1
%e A206627 ..2..0..1..1..0....1..1..1..0..2....2..2..2..0..2....2..1..1..1..1
%e A206627 ..1..1..1..1..1....1..2..1..1..1....1..1..1..1..0....2..0..1..2..2
%K A206627 nonn,new
%O A206627 1,1
%A A206627 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206621
%S A206621 14,86,86,580,1155,580,4035,17708,17708,4035,27895,270869,623933,
%T A206621 270869,27895,192358,4099700,21744156,21744156,4099700,192358,1327931,
%U A206621 62250328,752689692,1719525751,752689692,62250328,1327931,9168477,945314307
%N A206621 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206621 Table starts
%C A206621 ......14..........86............580..............4035.................27895
%C A206621 ......86........1155..........17708............270869...............4099700
%C A206621 .....580.......17708.........623933..........21744156.............752689692
%C A206621 ....4035......270869.......21744156........1719525751..........135010973857
%C A206621 ...27895.....4099700......752689692......135010973857........24021101375251
%C A206621 ..192358....62250328....26135187233....10633148505553......4289914143971114
%C A206621 .1327931...945314307...907138039901...837071761231340....765786718381692875
%C A206621 .9168477.14350695690.31481198946322.65890955765683798.136684817422844172008
%H A206621 R. H. Hardin, <a href="/A206621/b206621.txt">Table of n, a(n) for n = 1..84</a>
%e A206621 Some solutions for n=4 k=3
%e A206621 ..0..0..1..0....0..1..2..1....0..0..0..0....0..0..0..0....0..1..1..2
%e A206621 ..0..2..0..1....1..0..1..2....1..2..1..2....1..2..1..2....1..2..2..0
%e A206621 ..2..0..0..1....0..1..1..2....1..0..2..0....1..0..0..0....0..1..0..2
%e A206621 ..2..0..1..2....0..2..0..1....1..2..1..1....1..2..1..2....2..1..2..0
%e A206621 ..1..2..0..2....1..2..1..2....1..0..1..2....2..1..0..2....2..0..2..0
%K A206621 nonn,tabl,new
%O A206621 1,1
%A A206621 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206620
%S A206620 1327931,945314307,907138039901,837071761231340,765786718381692875,
%T A206620 704228581922250103909
%N A206620 Number of (n+1)X8 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206620 Column 7 of A206621
%e A206620 Some solutions for n=4
%e A206620 ..0..0..0..1..2..1..2..2....0..0..0..0..1..0..0..1....0..1..2..1..2..1..2..1
%e A206620 ..1..2..1..2..2..2..0..0....1..2..1..2..1..2..1..0....1..0..1..1..0..1..1..2
%e A206620 ..2..1..2..0..1..0..1..2....0..0..0..0..1..0..2..1....0..2..1..2..1..0..0..1
%e A206620 ..2..0..2..2..0..1..2..1....1..2..1..2..0..1..2..0....1..0..2..0..2..2..1..2
%e A206620 ..2..1..2..1..2..2..1..1....0..1..2..0..2..0..2..1....2..2..1..1..1..0..0..2
%K A206620 nonn,new
%O A206620 1,1
%A A206620 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206619
%S A206619 192358,62250328,26135187233,10633148505553,4289914143971114,
%T A206619 1738678950539792051,704228581922250103909,285197699391932223744199,
%U A206619 115508361295861948938141134,46781700029539016314279246512
%N A206619 Number of (n+1)X7 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206619 Column 6 of A206621
%H A206619 R. H. Hardin, <a href="/A206619/b206619.txt">Table of n, a(n) for n = 1..12</a>
%e A206619 Some solutions for n=4
%e A206619 ..0..0..1..1..0..0..1....0..1..0..2..1..2..2....0..0..0..0..0..0..1
%e A206619 ..0..2..0..2..0..1..2....1..1..0..1..2..0..0....2..1..2..1..2..1..2
%e A206619 ..1..0..2..1..1..2..2....2..0..1..2..0..0..1....0..0..2..0..2..0..1
%e A206619 ..2..2..1..0..0..1..0....1..2..0..1..2..2..0....2..1..0..1..1..1..0
%e A206619 ..1..2..1..0..1..2..0....0..1..1..2..0..1..2....1..0..1..2..1..2..1
%K A206619 nonn,new
%O A206619 1,1
%A A206619 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206618
%S A206618 27895,4099700,752689692,135010973857,24021101375251,4289914143971114,
%T A206618 765786718381692875,136684817422844172008,24398259565684468482150,
%U A206618 4355036966257280535391081,777364691410538473251350415
%N A206618 Number of (n+1)X6 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206618 Column 5 of A206621
%H A206618 R. H. Hardin, <a href="/A206618/b206618.txt">Table of n, a(n) for n = 1..210</a>
%e A206618 Some solutions for n=4
%e A206618 ..0..0..1..0..1..0....0..1..1..2..0..1....0..0..0..0..1..1....0..0..0..1..2..1
%e A206618 ..0..1..0..0..0..2....2..2..0..1..1..2....1..2..1..2..2..0....2..0..1..0..2..2
%e A206618 ..2..0..2..1..2..0....2..1..2..2..0..0....0..0..0..2..0..2....0..1..2..0..1..0
%e A206618 ..0..1..2..0..1..1....1..0..2..0..1..1....2..0..1..0..1..0....2..2..0..1..2..2
%e A206618 ..2..2..1..0..2..0....2..2..1..2..0..1....0..1..2..0..0..2....2..1..2..1..2..0
%K A206618 nonn,new
%O A206618 1,1
%A A206618 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206617
%S A206617 4035,270869,21744156,1719525751,135010973857,10633148505553,
%T A206617 837071761231340,65890955765683798,5187056339251009089,
%U A206617 408327809631129343522,32143691853804370071056,2530367382763227289642909
%N A206617 Number of (n+1)X5 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206617 Column 4 of A206621
%H A206617 R. H. Hardin, <a href="/A206617/b206617.txt">Table of n, a(n) for n = 1..210</a>
%e A206617 Some solutions for n=4
%e A206617 ..0..1..0..2..1....0..0..0..1..2....0..0..0..1..0....0..0..0..0..0
%e A206617 ..1..1..2..1..1....2..0..2..0..1....2..0..1..2..0....1..2..1..2..1
%e A206617 ..2..1..0..1..0....1..2..0..0..1....1..2..0..2..1....2..1..2..1..0
%e A206617 ..1..2..2..2..1....2..2..0..1..0....1..2..0..1..2....1..0..0..2..1
%e A206617 ..1..0..1..0..1....2..0..2..0..0....2..0..2..1..2....1..2..1..1..2
%K A206617 nonn,new
%O A206617 1,1
%A A206617 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206616
%S A206616 580,17708,623933,21744156,752689692,26135187233,907138039901,
%T A206616 31481198946322,1092605572922527,37920364043243138,
%U A206616 1316071917345653001,45675935634088583339,1585240798851880252784,55017771164461157130806
%N A206616 Number of (n+1)X4 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206616 Column 3 of A206621
%H A206616 R. H. Hardin, <a href="/A206616/b206616.txt">Table of n, a(n) for n = 1..210</a>
%F A206616 Empirical: a(n) = 37*a(n-1) -193*a(n-2) +4563*a(n-3) -26088*a(n-4) +213640*a(n-5) -2282905*a(n-6) +369531*a(n-7) +59365865*a(n-8) -119009270*a(n-9) -262257680*a(n-10) +527736231*a(n-11) +509956986*a(n-12) +4324710244*a(n-13) -23957378452*a(n-14) +23663119594*a(n-15) +61337666069*a(n-16) -218666604368*a(n-17) +228502952174*a(n-18) -85986086663*a(n-19) +669360026328*a(n-20) -249999620622*a(n-21) -2694789374595*a(n-22) +3166056347472*a(n-23) -4527450747937*a(n-24) +1962425462655*a(n-25) +11253630720214*a(n-26) -6950215731178*a(n-27) -8616642530280*a(n-28) +5352219023522*a(n-29) +18282316992761*a(n-30) +7753157593402*a(n-31) -40511908730324*a(n-32) +4939025872094*a(n-33) -60334135799883*a(n-34) +58664554195163*a(n-35) -121534792363268*a(n-36) +68593536567899*a(n-37) +27074083178241*a(n-38) +308051491547424*a(n-39) -39985515053935*a(n-40) +39295040533166*a(n-41) +83890931412882*a(n-42) -250291599920068*a(n-43) -288209619371358*a(n-44) -184438801479103*a(n-45) +38926434092468*a(n-46) -59238978126303*a(n-47) +106546892950000*a(n-48) +214149676500053*a(n-49) +116025062188772*a(n-50) -2859191597854*a(n-51) +39054500953935*a(n-52) -17011926676243*a(n-53) -44327153317680*a(n-54) -19261125626130*a(n-55) -3153801775828*a(n-56) -4604960888768*a(n-57) -7048617640872*a(n-58) -822479585408*a(n-59) +2623759339152*a(n-60) +177002043136*a(n-61) -414732129408*a(n-62) +52243793472*a(n-63) -4364697600*a(n-64) -4934155008*a(n-65) +884524032*a(n-66) -33288192*a(n-67)
%e A206616 Some solutions for n=4
%e A206616 ..0..0..0..0....0..0..1..2....0..1..2..1....0..0..1..2....0..1..0..2
%e A206616 ..1..2..1..2....0..1..0..0....1..0..1..2....2..1..2..2....1..2..2..1
%e A206616 ..1..0..0..0....2..0..2..1....0..1..1..2....1..0..0..1....1..0..1..2
%e A206616 ..1..2..1..2....1..0..2..0....0..2..0..1....1..2..1..2....0..1..2..0
%e A206616 ..2..1..0..2....0..1..0..1....1..2..1..2....2..0..2..0....0..2..0..0
%K A206616 nonn,new
%O A206616 1,1
%A A206616 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206615
%S A206615 86,1155,17708,270869,4099700,62250328,945314307,14350695690,
%T A206615 217872572862,3307774471061,50218674117508,762421824428498,
%U A206615 11575121642488239,175733960717293322,2668000123752680318
%N A206615 Number of (n+1)X3 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206615 Column 2 of A206621
%H A206615 R. H. Hardin, <a href="/A206615/b206615.txt">Table of n, a(n) for n = 1..210</a>
%F A206615 Empirical: a(n) = 14*a(n-1) +9*a(n-2) +199*a(n-3) -961*a(n-4) +12*a(n-5) -65*a(n-6) +4451*a(n-7) -2813*a(n-8) +623*a(n-9) -1152*a(n-10) +1104*a(n-11) -1036*a(n-12) +248*a(n-13) -408*a(n-14) +280*a(n-15)
%e A206615 Some solutions for n=4
%e A206615 ..0..0..1....0..0..0....0..0..1....0..0..0....0..0..0....0..1..0....0..0..0
%e A206615 ..2..1..2....1..2..1....2..0..1....1..0..1....1..2..1....2..1..2....1..0..1
%e A206615 ..0..1..2....1..0..2....0..1..2....2..1..0....0..2..0....2..0..1....0..2..0
%e A206615 ..1..2..0....0..1..0....1..1..2....0..1..1....0..1..2....0..1..1....0..0..2
%e A206615 ..0..1..0....2..0..2....0..0..1....2..0..1....1..0..1....0..1..0....0..1..0
%K A206615 nonn,new
%O A206615 1,1
%A A206615 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206614
%S A206614 14,86,580,4035,27895,192358,1327931,9168477,63292982,436934969,
%T A206614 3016371943,20823400074,143753294443,992394181981,6850947582710,
%U A206614 47295197965905,326500200032495,2253978967732298,15560239102262371
%N A206614 Number of (n+1)X2 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206614 Column 1 of A206621
%H A206614 R. H. Hardin, <a href="/A206614/b206614.txt">Table of n, a(n) for n = 1..210</a>
%F A206614 Empirical: a(n) = 7*a(n-1) -5*a(n-2) +36*a(n-3) -42*a(n-4) for n>5
%e A206614 Some solutions for n=4
%e A206614 ..0..0....0..0....0..0....0..0....0..0....0..1....0..0....0..1....0..0....0..0
%e A206614 ..1..2....1..1....0..1....0..1....0..1....2..2....0..1....2..0....0..1....1..2
%e A206614 ..2..1....1..2....1..2....2..0....1..2....2..1....1..2....1..2....1..2....2..1
%e A206614 ..2..2....2..0....0..2....2..2....0..2....0..2....2..2....2..0....1..1....2..0
%e A206614 ..1..0....1..1....0..1....0..0....2..0....2..2....2..1....0..1....2..0....2..1
%K A206614 nonn,new
%O A206614 1,1
%A A206614 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206613
%S A206613 14,1155,623933,1719525751,24021101375251,1738678950539792051
%N A206613 Number of (n+1)X(n+1) 0..2 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..2 introduced in row major order
%C A206613 Diagonal of A206621
%e A206613 Some solutions for n=4
%e A206613 ..0..0..1..2..2....0..0..0..0..1....0..0..0..1..2....0..0..1..2..2
%e A206613 ..2..1..2..1..1....1..2..1..2..0....1..0..2..0..2....2..1..0..0..1
%e A206613 ..1..2..1..1..2....1..0..2..1..0....0..1..0..0..1....2..0..1..1..0
%e A206613 ..1..1..0..2..1....1..2..0..0..2....2..0..2..2..0....0..2..2..0..1
%e A206613 ..2..0..1..2..1....0..0..1..0..1....1..2..1..2..1....0..1..0..2..1
%K A206613 nonn,new
%O A206613 1,1
%A A206613 R. H. Hardin (rhhardin(AT)att.net) Feb 10 2012

%I A206418
%S A206418 22,127,55,1527,18453,5517
%N A206418 The least integer k > 1 such that 1 + k^(5^n) + k^(2*5^n) + k^(3*5n) + k^(4*5^n) is prime.
%C A206418 Phi[5^(n+1),k]=1+k^(5^n)+k^(2*5^n)+k^(3*5^n)+k^(4*5^n)
%C A206418 The primes correspond to k(1) through k(4) have a p-1 factorable up to 34% or higher, thus proven by OpenPFGW.
%C A206418 The fifth one, Phi[5^6,18453]=1+18453^3125+18453^6250+18453^9375+18453^12500, is a 55,326 digits Fermat and Lucas PRP with 78.86% proof.  A CHG proofing is running but it will take month to complete.
%C A206418 The sixth one, Phi[5^7,5517], is 233,857 digits and can only be factored to about 26%.  It is too big for CHG to provide a proof.
%H A206418 David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/chgcertd.gp">Coppersmith--Howgrave-Graham certificate tester</a> (2006)
%H A206418 Chris Caldwell, <a href="http://primes.utm.edu/bios/page.php?id=797">John Renze's Coppersmith-Howgrave-Graham PARI script</a>
%e A206418 Phi[5^2,22] = 705429635566498619547944801 is prime, while Phi[25,k] with k = 2 to 21 are composites, so k(1)=22.
%t A206418 Table[i = 1; m = 5^u; While[i++; cp = 1 + i^m + i^(2*m) + i^(3*m) + i^(4^m); ! PrimeQ[cp]]; i, {u, 1, 4}]
%o A206418 (PARI) See Broadhurst link.
%o A206418 (PARI) a(n)=my(k=2);while(!ispseudoprime(polcyclo(5,k^n)),k++);k \\ Charles R Greathouse IV, Feb 09 2012
%Y A206418 Cf. A153438.
%K A206418 nonn,hard,new
%O A206418 1,1
%A A206418 Lei Zhou (lzhou5(AT)emory.edu), Feb 09 2012

%I A194709
%S A194709 30,15,15,6,10,14
%N A194709 Triangle read by rows: T(k,m) = number of ocurrences of k in the outer shell of the partitions of (9 + m).
%C A194709 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 9. For further information see A182703 and A135010.
%F A194709 T(k,m) = A182703(9+m,k), with T(k,m) = 0 if k > 9+m.
%F A194709 T(k,m) = A194812(9+m,k).
%e A194709 Triangle begins:
%e A194709 30,
%e A194709 15, 15,
%e A194709 6, 10, 14,
%e A194709 ...
%e A194709 For k = 1 and  m = 1; T(1,1) = 30 because there are 30 parts of size 1 in the outer shell of the partitions of 10, since 9 + m = 10, so a(1) = 30. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the outer shell of the partitions of 10, since 9 + m = 10, so a(2) = 15.
%Y A194709 Always the sum of row k = p(9) = A000041(n) = 30.
%Y A194709 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194708, this sequence, A194710.
%Y A194709 Cf. A135010, A138121, A194812.
%K A194709 nonn,tabl,more,new
%O A194709 1,1
%A A194709 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194708
%S A194708 22,7,15,6,6,10,2,5,5,10
%N A194708 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (8 + m).
%C A194708 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 8. For further information see A182703 and A135010.
%F A194708 T(k,m) = A182703(8+m,k), with T(k,m) = 0 if k > 8+m.
%F A194708 T(k,m) = A194812(8+m,k).
%e A194708 Triangle begins:
%e A194708 22,
%e A194708 7, 15,
%e A194708 6, 6, 10,
%e A194708 2, 5, 5, 10,
%e A194708 2, 3, 4, 5, 8,
%e A194708 ...
%e A194708 For k = 1 and  m = 1; T(1,1) = 22 because there are 22 parts of size 1 in the outer shell of the partitions of 9, since 8 + m = 9, so a(1) = 22. For k = 2 and m = 1; T(2,1) = 7 because there are seven parts of size 2 in the outer shell of the partitions of 9, since 8 + m = 9, so a(2) = 7.
%Y A194708 Always the sum of row k = p(8) = A000041(8) = 22.
%Y A194708 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194707, this sequence, A194709, A194710.
%Y A194708 Cf. A135010, A138121, A194812.
%K A194708 nonn,tabl,more,new
%O A194708 1,1
%A A194708 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194707
%S A194707 15,8,7,3,6,6,3,2,5,5
%N A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (7 + m).
%C A194707 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 7. For further information see A182703 and A135010.
%F A194707 T(k,m) = A182703(7+m,k), with T(k,m) = 0 if k > 7+m.
%F A194707 T(k,m) = A194812(7+m,k).
%e A194707 Triangle begins:
%e A194707 15,
%e A194707 8, 7,
%e A194707 3, 6, 6,
%e A194707 3, 2, 5, 5,
%e A194707 ...
%e A194707 For k = 1 and  m = 1; T(1,1) = 15 because there are 15 parts of size 1 in the outer shell of the partitions of 8, since 7 + m = 8, so a(1) = 15. For k = 2 and m = 1; T(2,1) = 8 because there are eight parts of size 2 in the outer shell of the partitions of 8, since 7 + m = 8, so a(2) = 8.
%Y A194707 Always the sum of row k = p(7) = A000041(7) = 15.
%Y A194707 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.
%Y A194707 Cf. A135010, A138121, A194812.
%K A194707 nonn,tabl,more,new
%O A194707 1,1
%A A194707 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194706
%S A194706 11,3,8,2,3,6,1,3,2,5,1,1,2,3,4,0,1,1,2,2,5
%N A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (6 + m).
%C A194706 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 6. For further information see A182703 and A135010.
%F A194706 T(k,m) = A182703(6+m,k), with T(k,m) = 0 if k > 6+m.
%F A194706 T(k,m) = A194812(6+m,k).
%e A194706 Triangle begins:
%e A194706 11,
%e A194706 3, 8,
%e A194706 2, 3, 6,
%e A194706 1, 3, 2, 5,
%e A194706 1, 1, 2, 3, 4,
%e A194706 0, 1, 1, 2, 2, 5,
%e A194706 ...
%e A194706 For k = 1 and  m = 1; T(1,1) = 11 because there are 11 parts of size 1 in the outer shell of the partitions of 7, since 6 + m = 7, so a(1) = 11. For k = 2 and m = 1; T(2,1) = 3 because there are three parts of size 2 in the outer shell of the partitions of 7, since 6 + m = 7, so a(2) = 3.
%Y A194706 Always the sum of row k = p(6) = A000041(6) = 11.
%Y A194706 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194705, this sequence, A194707-A194710.
%Y A194706 Cf. A135010, A138121, A194812.
%K A194706 nonn,tabl,more,new
%O A194706 1,1
%A A194706 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A185351
%S A185351 0,6,28,34,496,502,524,530,8128,8134,8156,8162,8624,8630,8652,8658,
%T A185351 33550336,33550342,33550364,33550370,33550832,33550838,33550860,
%U A185351 33550866,33558464,33558470,33558492,33558498,33558960,33558966,33558988,33558994,8589869056
%N A185351 Sums of distinct perfect numbers.
%C A185351 The first 131072 terms of this sequence are even. Conjecturally, all terms are even.
%H A185351 Charles R Greathouse IV, <a href="/A185351/b185351.txt">Table of n, a(n) for n = 1..10000</a>
%e A185351 502 = 496 + 6, where 496 and 6 are perfect.
%t A185351 With[{perf = Select[Range[10000], DivisorSigma[1, #] == 2# &]}, Rest[Union[Total/@Subsets[perf]]]] (* From Harvey P. Dale, Feb 07 2012 *)
%o A185351 (PARI) vecsum(v)=sum(i=1,#v,v[i]);
%o A185351 v=apply(n->binomial(n+1,2), select(k->ispseudoprime(k), vector(15,n,2^prime(n)-1))); u=List();for(i=0,2^#v-1,listput(u,vecsum(vecextract(v,i))));vecsort(Vec(u)) \\ Charles R Greathouse IV, Feb 09 2012
%Y A185351 Cf. A083865, A204879.
%K A185351 nonn,new
%O A185351 1,2
%A A185351 Charles R Greathouse IV (charles.greathouse(AT)case.edu), Feb 09 2012

%I A204594
%S A204594 13,4,0,3,6,10,13,17,20,24,28,32,36,40,44,49,53,57,62,66,71,76,80,
%T A204594 85,90,95,100,105,109,115,120,125,130,135,140,145,151,156,161,167,172,
%U A204594 177,183,188,194,199,205,210,216,222,227,233,239,244,250,256,262
%V A204594 -13,-4,0,3,6,10,13,17,20,24,28,32,36,40,44,49,53,57,62,66,71,76,80,
%W A204594 85,90,95,100,105,109,115,120,125,130,135,140,145,151,156,161,167,172,
%X A204594 177,183,188,194,199,205,210,216,222,227,233,239,244,250,256,262
%N A204594 Nearest integer to n*log(n) + n*log(log(n)) - n + n/log(n)*(log(log(n))-2) - n*log(log(n))*(log(log(n))-6)/(2 log(n)^2), an asymptotic expression for prime(n).
%H A204594 Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a> (2010).
%H A204594 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>
%t A204594 Table[Round[n Log[n] + n Log[Log[n]] - n + n/Log[n](Log[Log[n]] - 2) - n Log[Log[n]](Log[Log[n]] - 6)/(2 Log[n]^2)], {n, 2, 58}] (* Alonso del Arte, Feb 07 2012 *)
%o A204594 (PARI) A204594(n)=round(n*log(n)+n*log(log(n))-n+n/log(n)*(log(log(n))-2)-n*log(log(n))/2/log(n)^2*(log(log(n))-6))
%Y A204594 Cf. A059112, A059111, A064658, A064659.
%K A204594 sign,new
%O A204594 2,1
%A A204594 M. F. Hasler (oeis2012-removeThis(AT)hasler.fr), Feb 07 2012

%I A185342
%S A185342 2,4,4,6,12,8,8,24,32,16,10,40,80,80,32,12,60,160,240,192,64,
%T A185342 14,84,280,560,672,448,128,16,112,448,1120,1792,1792,1024,256,
%U A185342 18,144,672,2016,4032,5376,4608,2304,512,20,180,960,3360,8064
%V A185342 2,4,-4,6,-12,8,8,-24,32,-16,10,-40,80,-80,32,12,-60,160,-240,192,-64,
%W A185342 14,-84,280,-560,672,-448,128,16,-112,448,-1120,1792,-1792,1024,-256,
%X A185342 18,-144,672,-2016,4032,-5376,4608,-2304,512,20,-180,960,-3360,8064
%N A185342 Triangle of successive recurrences in columnns of A117317(n).
%C A185342 A117317 (A):
%C A185342 1
%C A185342 2    1
%C A185342 4    5   1
%C A185342 8   16   9   1
%C A185342 16  44  41  14   1
%C A185342 32 112 146  85  20  1
%C A185342 64 272 456 377 155 27 1
%C A185342 have for their columns successive signatures
%C A185342 (2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
%C A185342 Take based on abs(A133156) (B):
%C A185342 1
%C A185342 2   0
%C A185342 4   1  0
%C A185342 8   4  0 0
%C A185342 16 12  1 0 0
%C A185342 32 32  6 0 0 0
%C A185342 64 80 24 1 0 0 0.
%C A185342 The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
%C A185342 A053220 + A001787 = A014480.
%F A185342 Take A133156(n) without 1's or -1's double triangle (C)=
%F A185342 2
%F A185342 4
%F A185342 8      -4
%F A185342 16    -12
%F A185342 32    -32   6
%F A185342 64    -80  24
%F A185342 128  -192  80   -8
%F A185342 256  -448 240  -40
%F A185342 512 -1024 672 -160 10;
%F A185342 a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
%F A185342 Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
%e A185342 2                                         (1)
%e A185342 4    -4                                   (2)
%e A185342 6   -12   8                               (3)
%e A185342 8   -24  32   -16                         (4)
%e A185342 10  -40  80   -80   32                    (5)
%e A185342 12  -60 160  -240  192   -64              (6)
%e A185342 14  -84 280  -560  672  -448  128         (7)
%e A185342 16 -112 448 -1120 1792 -1792 1024 -256    (8)
%e A185342 Successive rows can be divided by A171977.
%Y A185342 Cf. For (A):A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group)). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.
%K A185342 sign,new
%O A185342 0,1
%A A185342 Paul Curtz (bpcrtz(AT)free.fr), Jan 26 2012

%I A205325
%S A205325 0,4,1,6,6,6,2
%N A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - ln(n) as n approaches infinity.
%H A205325 Martin Janecke, <a href="http://prlbr.de/2010/01/edle-reihe/">Edle Reihe</a>
%F A205325 lim_{n->infinity} (1 + 1/[2;2,...] + 1/[3;3,...] + ... + 1/[n;n,...] - ln(n)).
%F A205325 lim_{n->infinity} (sum_{k=1...n} (2/(k + sqrt(k^2 + 4))) - ln(n)).
%e A205325 0.0416662....
%Y A205325 Cf. A001620, A205326, continued fractions A001622, A014176, A098316, A098317, A098318.
%K A205325 cons,more,nonn,new
%O A205325 0,2
%A A205325 Martin Janecke (oeis.org(AT)prlbr.de), Jan 26 2012

%I A206612
%S A206612 13,6,19,25,44,69,113,182,295,477,772,1249,2021,3270,5291,8561,13852,
%T A206612 22413,36265,58678,94943,153621,248564,402185,650749,1052934,1703683,
%U A206612 2756617,4460300,7216917,11677217,18894134,30571351,49465485,80036836,129502321
%N A206612 Fibonacci sequence beginning 13 6.
%t A206612 LinearRecurrence[{1, 1}, {13, 6}, 80]
%Y A206612 Cf. A000032, A000045, A206610, A206611
%K A206612 nonn,new
%O A206612 1,1
%A A206612 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206611
%S A206611 13,7,20,27,47,74,121,195,316,511,827,1338,2165,3503,5668,9171,14839,
%T A206611 24010,38849,62859,101708,164567,266275,430842,697117,1127959,1825076,
%U A206611 2953035,4778111,7731146,12509257,20240403,32749660,52990063,85739723,138729786
%N A206611 Fibonacci sequence beginning 13 7.
%t A206611 LinearRecurrence[{1, 1}, {13, 7}, 80]
%Y A206611 Cf. A000032, A000045, A206609, A206610
%K A206611 nonn,new
%O A206611 1,1
%A A206611 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206610
%S A206610 13,8,21,29,50,79,129,208,337,545,882,1427,2309,3736,6045,9781,15826,
%T A206610 25607,41433,67040,108473,175513,283986,459499,743485,1202984,1946469,
%U A206610 3149453,5095922,8245375,13341297,21586672,34927969,56514641,91442610,147957251
%N A206610 Fibonacci sequence beginning 13 8.
%t A206610 LinearRecurrence[{1, 1}, {13, 8}, 80]
%Y A206610 Cf. A000032, A000045, A206608, A206609
%K A206610 nonn,new
%O A206610 1,1
%A A206610 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206609
%S A206609 13,9,22,31,53,84,137,221,358,579,937,1516,2453,3969,6422,10391,16813,
%T A206609 27204,44017,71221,115238,186459,301697,488156,789853,1278009,2067862,
%U A206609 3345871,5413733,8759604,14173337,22932941,37106278,60039219,97145497,157184716
%N A206609 Fibonacci sequence beginning 13 9.
%t A206609 LinearRecurrence[{1, 1}, {13, 9}, 80]
%Y A206609 Cf. A000032, A000045, A206607, A206608
%K A206609 nonn,new
%O A206609 1,1
%A A206609 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206608
%S A206608 13,10,23,33,56,89,145,234,379,613,992,1605,2597,4202,6799,11001,
%T A206608 17800,28801,46601,75402,122003,197405,319408,516813,836221,1353034,
%U A206608 2189255,3542289,5731544,9273833,15005377,24279210,39284587,63563797,102848384,166412181
%N A206608 Fibonacci sequence beginning 13 10.
%t A206608 LinearRecurrence[{1, 1}, {13, 10}, 80]
%Y A206608 Cf. A000032, A000045, A206607
%K A206608 nonn,new
%O A206608 1,1
%A A206608 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206607
%S A206607 13,11,24,35,59,94,153,247,400,647,1047,1694,2741,4435,7176,11611,
%T A206607 18787,30398,49185,79583,128768,208351,337119,545470,882589,1428059,
%U A206607 2310648,3738707,6049355,9788062,15837417,25625479,41462896,67088375,108551271,175639646
%N A206607 Fibonacci sequence beginning 13 11.
%t A206607 LinearRecurrence[{1, 1}, {13, 11}, 80]
%Y A206607 Cf. A000032, A000045
%K A206607 nonn,new
%O A206607 1,1
%A A206607 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206606
%S A206606 2089,4481,7057,15193,15641,16649,23417,34721,65537,68489,69697,72577,
%T A206606 93241,118673,123209,146161,173897,176401,191969,199873,205721,216233
%N A206606 Primes that can be written as a sum of a positive square and a positive cube in more than two ways.
%C A206606 A subset of these, {65537, 93241, 191969, ..} allows this representation in more than 3 ways.
%e A206606 2089 = 19^2+12^3 = 33^2+10^3 = 45^2+4^3
%t A206606 t={}; Do[Do[AppendTo[t,n^2+m^3],{n,300}],{m,300}]; t=Sort[t]; t3={}; Do[If[t[[n]]==t[[n+2]]&&PrimeQ[t[[n]]],AppendTo[t3,t[[n]]]],{n,Length[t]-2}]; t3; f1[l_]:=Module[{t={}},Do[If[l[[n]]!=l[[n+1]],AppendTo[t,l[[n]]]],{n,Length[l]-1}];t]; (*ExtractSingleTermsOnly*) f1[t3]
%Y A206606 Cf. A054402, A123364, A162930
%K A206606 nonn,new
%O A206606 1,1
%A A206606 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 10 2012

%I A206590
%S A206590 0,0,1,0,0,0,3,2,0,0,1,0,0,0,3,0,2,0,1,0,0,0,3,4,0,8,1,0,0,0,7,0,0,0,
%T A206590 5,0,0,0,3,0,0,0,1,2,0,0,3,6,4,0,1,0,8,0,3,0,0,0,1,0,0,2,15,0,0,0,1,0,
%U A206590 0,0,11,0,0,4,1,0,0,0,3,8,0,0,1,0,0,0,3,0,2,0,1,0,0,0,7,0,6,2
%N A206590 Number of solutions (n,k) of k^3=n^3 (mod n), where 1<=k<n.
%e A206590 8 divides exactly 3 of the numbers 8^3-k^3 for k=1,2,...,7, so that a(8)=2.
%t A206590 f[n_, k_] := If[Mod[n^3 - k^3, n] == 0, 1, 0];
%t A206590 t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
%t A206590 a[n_] := Count[Flatten[t[n]], 1]
%t A206590 Table[a[n], {n, 2, 120}] (* A206590 *)
%K A206590 nonn,new
%O A206590 2,7
%A A206590 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206589
%S A206589 1,0,2,1,2,1,1,1,1,0,3,1,2,1,2,0,2,0,2,1,2,1,1,0,0,1,1,0,4,1,2,2,2,1,
%T A206589 1,1,1,1,3,1,3,1,1,2,1,0,3,1,1,1,1,0,2,1,2,2,1,1,4,0,1,1,0,0,2,0,2,2,
%U A206589 3,0,4,1,2,2,1,1,3,1,2,1,2,1,3,1,3,2,3,1,3,0,1,0,2,1,2,0,2,0,2
%N A206589 Number of solutions (n,k) of p(k+1)=p(n+1) (mod n), where 1<=k<n.
%C A206589 Related to A206588, which includes differences p-2.
%e A206589 For k=1 to 5, the numbers p(7)-p(k+1) are 14,12,10,6,4, so that a(6)=2.
%t A206589 f[n_,k_]:=If[Mod[Prime[n+1]-Prime[k+1],n]==0,1,0];
%t A206589 t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
%t A206589 a[n_] := Count[Flatten[t[n]], 1]
%t A206589 Table[a[n], {n, 2, 120}]  (* A206589 *)
%Y A206589 Cf. A206588.
%K A206589 nonn,new
%O A206589 2,3
%A A206589 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206588
%S A206588 0,1,1,0,1,1,2,1,1,0,1,1,1,2,2,0,2,1,2,1,1,1,2,1,1,0,2,0,3,1,2,2,3,1,
%T A206588 3,1,1,2,2,1,3,1,3,2,2,1,3,1,3,2,2,1,2,1,1,1,1,1,2,0,1,1,0,1,2,1,1,2,
%U A206588 1,1,1,1,2,2,3,0,3,0,1,1,2,0,4,1,2,1,3,1,5,1,1,0,1,0,2,0,2,1,2
%N A206588 Number of solutions (n,k) of prime(k)=prime(n) (mod n), where 1<=k<n.
%e A206588 For k=1 to 7, the numbers p(8)-p(k) are 17,16,14,12,8,6,4, so that a(8)=2.
%t A206588 f[n_, k_] := If[Mod[Prime[n] - Prime[k], n] == 0, 1, 0];
%t A206588 t[n_] := Flatten[Table[f[n, k], {k, 1, n - 1}]]
%t A206588 a[n_] := Count[Flatten[t[n]], 1]
%t A206588 Table[a[n], {n, 2, 120}]  (* A206588 *)
%Y A206588 Cf. A206589.
%K A206588 nonn,new
%O A206588 2,7
%A A206588 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206567
%S A206567 1,0,0,0,1,0,1,0,0,1,0,0,1,0,0,0,1,1,1,1,0,1,0,1,2,1,0,1,0,0,0,1,1,0,
%T A206567 0,0,0,1,0,0,2,0,0,1,0,1,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,
%U A206567 0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,2,1,0,1,0,0,1,0,0,1,0,0,0,1,1
%N A206567 S(m,n)=(number of terms common to the base 3 expansions of m and n), a symmetric matrix by antidiagonals.
%e A206567 Northwest corner:
%e A206567 1 0 0 1 0 0 1 0 0 1 0 0 1
%e A206567 0 1 0 0 1 0 0 1 0 0 1 0 0
%e A206567 0 0 1 1 1 0 0 0 0 0 0 1 1
%e A206567 1 0 1 2 1 0 1 0 0 1 0 1 2
%e A206567 0 1 1 1 2 0 0 1 0 0 1 1 1
%e A206567 0 0 0 0 0 1 1 1 0 0 0 0 0
%e A206567 1 0 0 1 0 1 2 1 0 1 0 0 1
%e A206567 0 1 0 0 1 1 1 2 0 0 1 0 0
%e A206567 0 0 0 0 0 0 0 0 1 1 1 1 1
%e A206567 1 0 0 1 0 0 1 0 1 2 1 1 2
%e A206567 0 1 0 0 1 0 0 1 1 1 2 1 1
%e A206567 0 0 1 1 1 0 0 0 1 1 1 2 2
%e A206567 1 0 1 2 1 0 1 0 1 2 1 2 3
%e A206567 The above submatrix has only one 3; of course, every nonnegative integer occurs infinitely many times in the matrix.
%e A206567 4 = 3 + 1 and 13 = 3^2 + 3 + 1, so S(13,4)=2.
%t A206567 d[n_] := IntegerDigits[n, 3];
%t A206567 t[n_] := Reverse[Array[d, 100][[n]]]
%t A206567 s[n_, k_] := Position[t[n], k]
%t A206567 t[m_, n_] := Sum[Length[Intersection[s[m, k], s[n, k]]], {k, 1, 2}]
%t A206567 TableForm[Table[t[m, n], {m, 1, 24},
%t A206567   {n, 1, 24}]]  (* A206567 as a matrix *)
%t A206567 Flatten[Table[t[i, n + 1 - i], {n, 1, 24},
%t A206567   {i, 1, n}]]   (* A206568 as a sequence *)
%K A206567 nonn,tabl,base,new
%O A206567 1,25
%A A206567 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206566
%S A206566 0,1,1,0,0,0,1,0,1,1,0,1,1,1,1,1,1,2,1,2,2,0,0,0,0,0,0,0,1,0,1,0,1,0,
%T A206566 1,1,0,1,1,0,0,1,1,1,1,1,1,2,0,1,1,2,1,2,2,0,0,0,1,1,1,1,1,1,1,1,1,0,
%U A206566 1,1,2,1,2,1,2,1,2,2,0,1,1,1,1,2,2,1,1,2,2,2,2,1,1,2,1,2,2,3,1
%N A206566 Triangular array:  T(i,j) = number of terms common to the binary expansions of i+1 and j, for j=1,2,3,...,i; i=1,2,3,...
%C A206566 Row n consists of the first n terms of column n+1 of A206479, and (nth row sum)=A115478(n+1).
%e A206566 First ten rows:
%e A206566 0
%e A206566 1 1
%e A206566 0 0 0
%e A206566 1 0 1 1
%e A206566 0 1 1 1 1
%e A206566 1 1 2 1 2 2
%e A206566 0 0 0 0 0 0 0
%e A206566 1 0 1 0 1 0 1 1
%e A206566 0 1 1 0 0 1 1 1 1
%e A206566 1 1 2 0 1 1 2 1 2 2
%t A206566 d[n_] := IntegerDigits[n, 2];
%t A206566 t[n_] := Reverse[Array[d, 120][[n]]]
%t A206566 s[n_] := Position[t[n], 1]
%t A206566 t[m_, n_] := Length[Intersection[s[m], s[n]]]
%t A206566 TableForm[Table[t[m, n], {m, 1, 14},
%t A206566   {n, 1, 14}]] (* A206479 as a matrix *)
%t A206566 Flatten[Table[t[i, n + 1 - i], {n, 1, 14},
%t A206566   {i, 1, n}]]  (* A206479 as a sequence *)
%t A206566 u = Table[t[n - 1, m], {n, 3, 16}, {m, 1, n - 2}];
%t A206566 TableForm[u]   (* A206566 as a triangle *)
%t A206566 Flatten[u]     (* A206566 as a sequence *)
%t A206566 v[n_] := Table[t[k, n + 1], {k, 1, n}]
%t A206566 Table[Count[v[n], 0], {n, 1, 100}] (* A115478 *)
%Y A206566 Cf. A206479, A115478.
%K A206566 nonn,tabl,base,new
%O A206566 1,18
%A A206566 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206479
%S A206479 1,0,0,1,1,1,0,1,1,0,1,0,2,0,1,0,0,0,0,0,0,1,1,1,1,1,1,1,0,1,1,1,1,1,
%T A206479 1,0,1,0,2,1,2,1,2,0,1,0,0,0,1,1,1,1,0,0,0,1,1,1,0,2,2,2,0,1,1,1,0,1,
%U A206479 1,0,0,2,2,0,0,1,1,0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,0,0,0,0,0,0,0
%N A206479 Number of terms common to the binary expansions of m and n; a matrix by antidiagonals.
%e A206479 Northwest corner (the antidiagonals can be read either
%e A206479 southwest or northeast, since the matrix is symmetric):
%e A206479 1 0 1 0 1 0 1 0 1 0
%e A206479 0 1 1 0 0 1 1 0 0 1
%e A206479 1 1 2 0 1 1 2 0 1 1
%e A206479 0 0 0 1 1 1 1 0 0 0
%e A206479 1 0 1 1 2 1 2 0 1 0
%e A206479 0 1 1 1 1 2 2 0 0 1
%e A206479 1 1 2 1 2 2 3 0 1 1
%e A206479 ...
%e A206479 11 = 1 + 1*2 + 1*8 and 29 = 1 + 1*4 + 1*8 + 1*16,
%e A206479 so that T(11,29)=2.
%t A206479 d[n_] := IntegerDigits[n, 2];
%t A206479 t[n_] := Reverse[Array[d, 120][[n]]]
%t A206479 s[n_] := Position[t[n], 1]
%t A206479 t[m_, n_] := Length[Intersection[s[m], s[n]]]
%t A206479 TableForm[Table[t[m, n], {m, 1, 14},
%t A206479   {n, 1, 14}]]  (* A206479 as a matrix *)
%t A206479 Flatten[Table[t[i, n + 1 - i], {n, 1, 14},
%t A206479   {i, 1, n}]]   (* A206479 as a sequence *)
%t A206479 u = Table[t[n - 1, m], {n, 3, 16}, {m, 1, n - 2}];
%t A206479 TableForm[u]    (* A206566 as a triangle *)
%t A206479 Flatten[u]      (* A206566 as a sequence *)
%t A206479 v[n_] := Table[t[k, n + 1], {k, 1, n}]
%t A206479 Table[Count[v[n], 0], {n, 1, 100}]  (* A115478 *)
%Y A206479 Cf. A206566, A115478.
%K A206479 nonn,tabl,base,new
%O A206479 1,13
%A A206479 Clark Kimberling (ck6(AT)evansville.edu), Feb 09 2012

%I A206446
%S A206446 2,6,7,9,10,16,18,21,22,23,26,28,29,30,32,33,34,42,45,49,50,52,56,58,
%T A206446 60,61,64,65,66,67,71,74,75,77,79,81,82,83,84,87,88,90,91,92,93,95,96,
%U A206446 97,98,108,112,117,118,121,126,128,131,132,136,137,138,140,145
%N A206446 Positions of 1 in A076478; complement of A206445.
%e A206446 The positions of 1 in 0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,...
%e A206446 are 2,6,7,9,10,16,...
%t A206446 d[n_] := Rest@IntegerDigits[n + 1, 2] + 1;
%t A206446 t = -1 + Flatten[Array[d, 100]]
%t A206446 Flatten[Position[t, 0]]  (* A206445 *)
%t A206446 Flatten[Position[t, 1]]  (* A206446 *)
%Y A206446 Cf. A076478, A206445.
%K A206446 nonn,new
%O A206446 1,1
%A A206446 Clark Kimberling (ck6(AT)evansville.edu), Feb 08 2012

%I A206445
%S A206445 1,3,4,5,8,11,12,13,14,15,17,19,20,24,25,27,31,35,36,37,38,39,40,41,
%T A206445 43,44,46,47,48,51,53,54,55,57,59,62,63,68,69,70,72,73,76,78,80,85,86,
%U A206445 89,94,99,100,101,102,103,104,105,106,107,109,110,111,113,114
%N A206445 Positions of 0 in A076478; complement of A206446.
%e A206445 The positions of 0 in 0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,...
%e A206445 are 1,3,4,5,9,11,12,13,14,15,...
%t A206445 d[n_] := Rest@IntegerDigits[n + 1, 2] + 1;
%t A206445 t = -1 + Flatten[Array[d, 100]]
%t A206445 Flatten[Position[t, 0]]  (* A206445 *)
%t A206445 Flatten[Position[t, 1]]  (* A206446 *)
%Y A206445 Cf. A076478, A206446.
%K A206445 nonn,new
%O A206445 1,2
%A A206445 Clark Kimberling (ck6(AT)evansville.edu), Feb 08 2012

%I A206444
%S A206444 13,53,213,853,3413,13653,54613
%N A206444 Least n such that L(n)<-1 and L(n)<L(n-1), where L(k) means the least root of the polynomial p(k,x) defined at A206284, and a(1)=13.
%C A206444 A206284 gives an ordering {p(n,x)} of the polynomials with coefficients in {0,1}.  The least n for which p(n,x) has a root r less than -1 is 13, hence the choice of 13 as the initial term of A206443.  (Specifically, p(13,x)=1+x^2+x^3, and r=-1.46557...)  The next p(n,x) having a root less than -1 and <r is p(53,x)=1+x^2+x^4+x^5, with least root -1.57014...
%C A206444 The first 7 terms of A206444 are also the first 7 terms of A072197 and A206371.
%t A206444 highs := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[Split[Rest[FoldList[Max, -\[Infinity], #]]]] &
%t A206444 f[polyInX_] := {Min[#], Max[#]} &[
%t A206444   Map[#[[1]] &, DeleteCases[Map[{#, Head[#]} &, Chop[N[x /. Solve[polyInX == 0, x], 40]]], {_, Complex}]]]
%t A206444 t = Table[IntegerDigits[n, 2], {n, 1, 100000}];
%t A206444 b[n_] := Reverse[Array[x^(# - 1) &, {n + 1}]]
%t A206444 p[n_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206444 Table[p[n], {n, 1, 25}]
%t A206444 fitCriterion = Intersection[Map[#[[1]] &, DeleteCases[
%t A206444        Table[{n, Boole[IrreduciblePolynomialQ[p[n]]]}, {n, 1, #}], {_, 0}]], Map[#[[1]] &, DeleteCases[
%t A206444        Table[{n, CountRoots[#, {x, -Infinity, 0}] -
%t A206444        CountRoots[#, {x, -1, 0}] &[p[n]]}, {n, 1, #}],
%t A206444            {_, 0}]]] &[Length[t]];
%t A206444 polyNum = Map[{f[p[#]][[1]], #} &, fitCriterion];
%t A206444 up = Map[polyNum[[#]] &, highs[Map[#[[1]] &, polyNum]][[2]]]
%t A206444 down = Map[polyNum[[#]] &, highs[Map[#[[1]] &, -polyNum]][[2]]]
%t A206444 Table[up[[k, 2]], {k, 1, Length[up]}]      (* A206443 *)
%t A206444 Table[down[[k, 2]], {k, 1, Length[down]}]  (* A206444 *)
%t A206444 (* Peter Moses, 6 Feb 2012 *)
%Y A206444 Cf. A206284, A206443.
%K A206444 nonn,new
%O A206444 1,1
%A A206444 Clark Kimberling (ck6(AT)evansville.edu), Feb 07 2012

%I A206443
%S A206443 13,37,145,157,181,517,565,661,2101,2197,2581,2773,8725,8917,10357,
%T A206443 10453,10837,35029,35413,41173,41557,43093,43861
%N A206443 Least n such that L(n)<-1 and L(n)>L(n-1), where L(k) means the least root of the polynomial p(k,x) defined at A206284, and a(1)=13.
%C A206443 A206284 gives an ordering {p(n,x)} of the polynomials
%C A206443  with coefficients in {0,1}.  The least n for which p(n,x) has a root r less than -1 is 13, hence the choice of 13 as the initial term of A206443.  (Specifically, p(13,x)=1+x^2+x^3, and r=-1.46557...)  The next p(n,x) having a root less than -1 and >r is p(37,x)=1+x^2+x^5, with least root -1.1938...
%t A206443 highs := {First /@ #, Most[FoldList[Plus, 1, Length /@ #]]} &[Split[Rest[FoldList[Max, -\[Infinity], #]]]] &
%t A206443 f[polyInX_] := {Min[#], Max[#]} &[
%t A206443   Map[#[[1]] &, DeleteCases[Map[{#, Head[#]} &, Chop[N[x /. Solve[polyInX == 0, x], 40]]], {_, Complex}]]]
%t A206443 t = Table[IntegerDigits[n, 2], {n, 1, 100000}];
%t A206443 b[n_] := Reverse[Array[x^(# - 1) &, {n + 1}]]
%t A206443 p[n_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206443 Table[p[n], {n, 1, 25}]
%t A206443 fitCriterion = Intersection[Map[#[[1]] &, DeleteCases[
%t A206443        Table[{n, Boole[IrreduciblePolynomialQ[p[n]]]}, {n, 1, #}], {_, 0}]], Map[#[[1]] &, DeleteCases[
%t A206443        Table[{n, CountRoots[#, {x, -Infinity, 0}] -
%t A206443        CountRoots[#, {x, -1, 0}] &[p[n]]}, {n, 1, #}],
%t A206443            {_, 0}]]] &[Length[t]];
%t A206443 polyNum = Map[{f[p[#]][[1]], #} &, fitCriterion];
%t A206443 up = Map[polyNum[[#]] &, highs[Map[#[[1]] &, polyNum]][[2]]]
%t A206443 down = Map[polyNum[[#]] &, highs[Map[#[[1]] &, -polyNum]][[2]]]
%t A206443 Table[up[[k, 2]], {k, 1, Length[up]}]      (* A206443 *)
%t A206443 Table[down[[k, 2]], {k, 1, Length[down]}]  (* A206444 *)
%t A206443 (* Peter Moses, 6 Feb 2012 *)
%Y A206443 Cf. A206284, A206444.
%K A206443 nonn,new
%O A206443 1,1
%A A206443 Clark Kimberling (ck6(AT)evansville.edu), Feb 07 2012

%I A206442
%S A206442 0,0,1,0,1,1,1,0,1,1,1,1,1,2,2,0,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,0,3,2,
%T A206442 2,1,1,2,2,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,2,1,3,3,1,1,1,2,2,0,3,1,1,1,
%U A206442 3,1,1,1,1,2,2,1,2,2,1,1,1,2,1,2,2,2,2,1,1,1,2,1,4,2,3,1,1,1,2
%N A206442 Number of distinct nonconstant factors of the polynomials p(n,x) defined at A206284.
%C A206442 The factorization is over the ring of polynomials having integer coefficients.
%e A206442 p(1,x)=0, so a(1)=0
%e A206442 p(2,x)=1, so a(2)=0
%e A206442 p(3,x)=x, so a(3)=1
%e A206442 p(11,x)=x^4, so a(4)=1
%e A206442 p(14,x)=1+x^3, so a(14)=2
%e A206442 p(33,x)=x+x^4, so a(33)=3
%e A206442 p(93,x)=x+x^10, so a(93)=4
%e A206442 p(177,x)=x+x^16, so a(177)=5
%t A206442 b[n_] := Table[x^k, {k, 0, n}];
%t A206442 f[n_] := f[n] = FactorInteger[n]; z = 1000;
%t A206442 t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]] == Prime[k], f[n][[m, 2]], 0];
%t A206442 u = Table[Apply[Plus,
%t A206442     Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
%t A206442       Length[f[n]]}]], {n, 1, z}];
%t A206442 p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
%t A206442 TableForm[Table[{n, FactorInteger[n],
%t A206442    p[n, x], -1 + Length[FactorList[p[n, x]]]},
%t A206442  {n, 1, z/4}]]
%t A206442 Table[-1 + Length[FactorList[p[n, x]]], {n, 1, z/4}]
%t A206442  (* A206442 *)
%Y A206442 Cf. A206284, A206285.
%K A206442 nonn,new
%O A206442 1,14
%A A206442 Clark Kimberling (ck6(AT)evansville.edu), Feb 07 2012

%I A206350
%S A206350 1,2,4,8,12,20,24,36,44,56,64,84,92,116,128,144,160,192,204,240,256,
%T A206350 280,300,344,360,400,424,460,484,540,556,616,648,688,720,768,792,864,
%U A206350 900,948,980,1060,1084,1168,1208,1256,1300,1392,1424,1508,1548
%N A206350 Position of 1/n in the canonical bijection from the positive integers to the positive rational numbers.
%C A206350 The canonical bijection from the positive integers to the positive rational numbers is given by A038568(n)/(A038569(n).
%e A206350 The canonical bijection starts with
%e A206350 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, so that A206297 starts with 1,3,5,9,13 and
%e A206350 A206350 starts with 1,2,4,8,12.
%t A206350 a[n_] := Module[{s = 1, k = 2, j = 1},
%t A206350   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
%t A206350   s = s - 2*EulerPhi[k - 1];
%t A206350   While[s <= n, If[GCD[j, k - 1] == 1,
%t A206350     s = s + 2]; j = j + 1];
%t A206350   If[s > n + 1, j - 1, k - 1]]; t =
%t A206350  Table[a[n], {n, 0, 3000}];   (* A038568 *)
%t A206350 ReplacePart[Flatten[Position[t, 1]], 1, 1] (* A206350 *)
%Y A206350 Cf. A038568, A038569, A206296.
%K A206350 nonn,new
%O A206350 1,2
%A A206350 Clark Kimberling (ck6(AT)evansville.edu), Feb 06 2012

%I A206331
%S A206331 0,1,2,7,8,11,12,13,14,15,16,23,24,25,26,27,28,31,32,35,36,49,50,51,
%T A206331 52,61,62,63,64,65,66,67,68,71,72,75,76,79,80,85,86,89,90,91,92,93,94,
%U A206331 95,96,99,100,103,104,107,108,111,112,115,116,121,122,127,128
%N A206331 Numbers that match polynomials not irreducible over the integers.
%C A206331 See the Comments and example at A206330.
%t A206331 (See the program at A206330.)
%Y A206331 Cf. A206330.
%K A206331 nonn,new
%O A206331 0,3
%A A206331 Clark Kimberling (ck6(AT)evansville.edu), Feb 06 2012

%I A206330
%S A206330 3,4,5,6,9,10,17,18,19,20,21,22,29,30,33,34,37,38,39,40,41,42,43,44,
%T A206330 45,46,47,48,53,54,55,56,57,58,59,60,69,70,73,74,77,78,81,82,83,84,87,
%U A206330 88,97,98,101,102,105,106,109,110,113,114,117,118,119,120,123
%N A206330 Numbers that match polynomials irreducible over the integers.
%C A206330 Each n>1 matches a polynomial having integer coefficients
%C A206330 determined by the prime factorization of n.  Let c be a
%C A206330 positive integer, and write
%C A206330 c=p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k), and
%C A206330 define p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
%C A206330 If c/d is a rational number with GCD(c,d)=1, define
%C A206330 Q(c/d,x)=p(c,x)-p(d,x).  Let c(n)/d(n) be the nth
%C A206330 positive rational number given by the canonical
%C A206330 bijection; i.e., c(n)=A038568(n)/A038569(n).
%C A206330 Define P(0,x)=1 and P(n,x)=Q(c(n)/d(n),x).  Polynomials
%C A206330 having nonnegative integer coefficients are matched to
%C A206330 the nonnegative integers as follows:
%C A206330 ...
%C A206330 n .... P[n,x] .. irreducible
%C A206330 0 .... 0 ....... no
%C A206330 1 ... -1 ....... no
%C A206330 2 .... 1 ....... no
%C A206330 3 ... -x ....... yes
%C A206330 4 .... x ....... yes
%C A206330 5 ... 1-x ...... yes
%C A206330 6 .. -1+x ...... yes
%C A206330 7 .. -2 ........ no
%C A206330 8 ... 2 ........ no
%C A206330 9 .. -2+x ...... yes
%C A206330 10 .. 2-x ...... yes
%e A206330 In the table under Comments, read "yes" for n=3,4,5,6,9,10.
%t A206330 b[n_] := Table[x^k, {k, 0, n}];
%t A206330 f[n_] := f[n] = FactorInteger[n]; z = 1000;
%t A206330 t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
%t A206330  == Prime[k], f[n][[m, 2]], 0];
%t A206330 u = Table[Apply[Plus,
%t A206330     Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
%t A206330       Length[f[n]]}]], {n, 1, z}];
%t A206330 c[n_] := Module[{s = 1, k = 2, j = 1},
%t A206330    While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
%t A206330    s = s - 2*EulerPhi[k - 1];
%t A206330    While[s <= n, If[GCD[j, k - 1]
%t A206330       == 1, s = s + 2]; j = j + 1];
%t A206330    If[s > n + 1, j - 1, k - 1]];
%t A206330 d[n_] := Module[{s = 1, k = 2, j = 1},
%t A206330    While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
%t A206330    s = s - 2*EulerPhi[k - 1];
%t A206330    While[s <= n, If[GCD[j, k - 1]
%t A206330       == 1, s = s + 2]; j = j + 1];
%t A206330    If[s > n + 1, k - 1, j - 1]];
%t A206330 P[n_, x_] :=
%t A206330  u[[c[n]]].b[-1 + Length[u[[c[n]]]]] -
%t A206330   u[[d[n]]].b[-1 + Length[u[[d[n]]]]]
%t A206330 TableForm[Table[{n, P[n, x], Factor[P[n, x]]},
%t A206330    {n, 1, z/4}]];
%t A206330 v = {}; Do[n++;
%t A206330  If[IrreduciblePolynomialQ[P[n, x]], AppendTo[v, n]], {n, z/2}]
%t A206330 v                            (* A206330 *)
%t A206330 Complement[Range[0,200], v]  (* A206331 *)
%Y A206330 Cf. A206284 (polynomials over the positive integers),
%Y A206330 A206331 (complement of A206330).
%K A206330 nonn,new
%O A206330 1,1
%A A206330 Clark Kimberling (ck6(AT)evansville.edu), Feb 06 2012

%I A206297
%S A206297 1,3,5,9,13,21,25,37,45,57,65,85,93,117,129,145,161,193,205,241,257,
%T A206297 281,301,345,361,401,425,461,485,541,557,617,649,689,721,769,793,865,
%U A206297 901,949,981,1061,1085,1169,1209,1257,1301,1393,1425,1509,1549
%N A206297 Position of n in the canonical bijection from the positive integers to the positive rational numbers.
%C A206297 The canonical bijection from the positive integers to the positive rational numbers is given by A038568(n)/(A038569(n).
%e A206297 The canonical bijection starts with
%e A206297 1/1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 3/4, 4/3, 1/5, 5/1, so that A206297 starts with 1,3,5,9,13 and
%e A206297 A206350 starts with 1,2,4,8,12.
%t A206297 a[n_] := Module[{s = 1, k = 2, j = 1},
%t A206297   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
%t A206297   s = s - 2*EulerPhi[k - 1];
%t A206297   While[s <= n, If[GCD[j, k - 1] =
%t A206297     = 1, s = s + 2]; j = j + 1];
%t A206297   If[s > n + 1, j - 1, k - 1]];
%t A206297  t = Table[a[n], {n, 0, 3000}];   (* A038568 *)
%t A206297 ReplacePart[1 + Flatten[Position[t, 1]], 1, 1]
%t A206297  (* A206297 *)
%Y A206297 Cf. A038568, A038569, A206350.
%K A206297 nonn,new
%O A206297 1,2
%A A206297 Clark Kimberling (ck6(AT)evansville.edu), Feb 06 2012

%I A195017
%S A195017 0,1,1,2,1,0,1,3,2,2,1,1,1,0,0,4,1,1,1,3,2,2,1,2,2,0,3,1,1,1,
%T A195017 1,5,0,2,0,0,1,0,2,4,1,1,1,3,1,2,1,3,2,3,0,1,1,2,2,2,2,0,1,2,
%U A195017 1,2,3,6,0,1,1,3,0,1,1,1,1,0,1,1,0,1,1,5,4,2,1,0,2,0
%V A195017 0,1,-1,2,1,0,-1,3,-2,2,1,1,-1,0,0,4,1,-1,-1,3,-2,2,1,2,2,0,-3,1,-1,1,
%W A195017 1,5,0,2,0,0,-1,0,-2,4,1,-1,-1,3,-1,2,1,3,-2,3,0,1,-1,-2,2,2,-2,0,1,2,
%X A195017 -1,2,-3,6,0,1,1,3,0,1,-1,1,1,0,1,1,0,-1,-1,5,-4,2,1,0,2,0
%N A195017 p(n,-1), where p(n,x) is defined at A206284.
%C A195017 Positions of 0 give the values of n for which the polynomial p(n,x) is divisible by x+1. For related sequences, see the Mathematica section.
%e A195017 The sequence can be read from a list of the polynomials:
%e A195017 p(1,x)=0
%e A195017 p(2,x)=1
%e A195017 p(3,x)=x
%e A195017 p(4,x)=2
%e A195017 p(5,x)=x^2
%e A195017 p(6,x)=1+x
%e A195017 p(7,x)=x^3.
%e A195017 (The extended list consists of all polynomials whose coefficients are nonnegative integers.)
%t A195017 b[n_] := Table[x^k, {k, 0, n}];
%t A195017 f[n_] := f[n] = FactorInteger[n]; z = 200;
%t A195017 t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
%t A195017 == Prime[k], f[n][[m, 2]], 0];
%t A195017 u = Table[Apply[Plus,
%t A195017     Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
%t A195017       Length[f[n]]}]], {n, 1, z}];
%t A195017 p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
%t A195017 Table[p[n, x] /. x -> 0, {n, 1, z/2}]   (* A007814 *)
%t A195017 Table[p[2 n, x] /. x -> 0, {n, 1, z/2}] (* A001511 *)
%t A195017 Table[p[n, x] /. x -> 1, {n, 1, z}]     (* A001222 *)
%t A195017 Table[p[n, x] /. x -> 2, {n, 1, z}]     (* A048675 *)
%t A195017 Table[p[n, x] /. x -> 3, {n, 1, z}]     (* A090880 *)
%t A195017 Table[p[n, x] /. x -> -1, {n, 1, z}]    (* A195017 *)
%Y A195017 sign
%K A195017 sign,new
%O A195017 1,4
%A A195017 Clark Kimberling (ck6(AT)evansville.edu), Feb 06 2012

%I A206547
%S A206547 1,5,11,13,17,19,23,25,29,31,37,41,43,47,53,55,59,61,65,67,71,73,79,
%T A206547 83,85,89,95,97,101,103,107,109,113,115,121,125,127,131,137,139,143,
%U A206547 145,149,151,155,157,163,167,169,173,179,181,185,187,191,193,197,199,205,209,211
%N A206547 Positive odd numbers relatively prime to 21.
%C A206547 These are the positive integers not divisible by 2,3, or 7.
%H A206547 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A206547 a(n) = a(n-12) + 42, n>=13.
%F A206547 a(n) = a(n-1) + a(n-12) - a(n-13), n>=13, with a(0)=-1.
%F A206547 a(n) = 2*n-1 + 2*sum(F21[j]*floor((n+(j-1))/12),j=1..12), with F21=[1,2,0,1,0,1,0,1,0,2,1,0], n>=1. For n=0 this becomes -1, but the following o.g.f. has a(0)=0 if it starts with x^0.
%F A206547 O.g.f.: x*(1+x^12+4*x*(1+x^10)+6*x^2*(1+x^8)+2*x^3*(1+x^6)+4*x^4*(1+x^4)+2*x^5*(1+x^2)+4*x^6)/((1-x^12)*(1-x)). The denominator could be factored into cyclotomic polynomials. Compare with the formula contribution from R. J. Mathar in A007775.
%Y A206547 Cf. A007775, A204458.
%K A206547 nonn,easy,new
%O A206547 1,2
%A A206547 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 10 2012

%I A206601
%S A206601 0,2,26,728,59048,14348906,10460353202,22876792454960,
%T A206601 150094635296999120,2954312706550833698642,
%U A206601 174449211009120179071170506,30903154382632612361920641803528,16423203268260658146231467800709255288,26183890704263137277674192438430182020124346
%N A206601 3^(n(n+1)/2) - 1.
%C A206601 There are n cities located on the vertices of a convex n-gon and 2 types of communication lines available. Any city can be connected to any other by only one communication line (that can be of any type). A network exists if at least 2 cities are connected by a communication line. The sequence shows how many different networks a(n) can be built. In general, if the number of communication-line types is c, then a(n) = (c+1)^(n(n+1)/2)-1. Thus other sequences of this type can be generated.
%F A206601 a(n) = (3^A000217)-1.
%e A206601 In the case of 2 different types of communication lines and 4 cities, the number of different networks (connecting at least 2 cities) is 728.
%Y A206601 cf. A000217, A126883.
%K A206601 easy,nonn,new
%O A206601 0,2
%A A206601 Ivan N. Ianakiev (ianakiev.ivan(AT)gmail.com), Feb 10 2012

%I A206548
%S A206548 1,5,11,13,17,19,19,17,13,11,5,1,1,5,11,13,17,19,19,17,13,11,5,1,1,5,
%T A206548 11,13,17,19,19,17,13,11,5,1,1,5,11,13,17,19,19,17,13,11,5,1,1,5,11,
%U A206548 13,17,19,19,17,13,11,5,1,1,5,11,13,17,19,19,17,13,11,5,1
%N A206548 Period 12: repeat 1, 5, 11, 13, 17, 19, 19, 17, 13, 11, 5, 1.
%C A206548 For general Mod n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Mod 21 of the positive odd integers relatively prime to 21 which are shown in A206547. The underlying periodic sequence with period length 42 is, with offset 0, called P_21 or also Mod21: [seq(j,j=0..20),0,seq(21-j,j=1..20)].
%F A206548 a(n) = A206547(n) (M0d 21) := Mod21(A206547(n)), n>=1, with the periodic sequence Mod21 (period length 42) given in the comment section.
%F A206548 O.g.f: x*(1+x)*(1+x+x^2)*(1+3*x+3*x^2+8*x^4+3*x^6+3*x^7+x^8)/ (1-x^8). The denominator could be factored into cyclotomic polynomials.
%e A206548 Residues Mod 21 of the positive odd integers relatively prime to 21:
%e A206548 A206547: 1, 5, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, ...
%e A206548 Mod 21:  1, 5, 11, 13, 17, 19, 19, 17, 13, 11,  5,  1,  1, ...
%Y A206548 Cf. A206546.
%K A206548 nonn,easy,new
%O A206548 1,2
%A A206548 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 10 2012

%I A206546
%S A206546 1,7,11,13,13,11,7,1,1,7,11,13,13,11,7,1,1,7,11,13,13,11,7,1,1,7,11,
%T A206546 13,13,11,7,1,1,7,11,13,13,11,7,1,1,7,11,13,13,11,7,1,1,7,11,13,13,11,
%U A206546 7,1
%N A206546 Period 8: repeat 1, 7, 11, 13, 13, 11, 7, 1.
%C A206546 For general Mod n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Mod 15 of the positive odd numbers relatively prime to 15 (the positive odd numbers from all reduced residue classes mod 15), shown in A007775. The underlying periodic sequence with period length 30 is [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,0,14,13,12,11,10,9,8,7,6,5,4,3,2,1], called, with offset 0, P_15 or Mod15.
%F A206546 a(n) = A007775(n) (Mod 15) := Mod15(A007775(n)), n>=1, with the periodic sequence Mod15 (period length 30) given in the comment section.
%F A206546 O.g.f: x*(1+x^7+7*x*(1+x^5)+11*x^2*(1+x^3)+13*x^3*(1+x))/(1-x^8) = x*(1+x)*(1+6*x+5*x^2+8*x^3+5*x^4+6*x^5+x^6)/(1-x^8).
%e A206546 Residues Mod 15 of the positive odd numbers relatively prime to 15:
%e A206546 A007775: 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, ...
%e A206546 Mod 15:  1, 7, 11, 13, 13, 11,  7,  1,  1,  7, 11, 13, 13, 11, ...
%Y A206546 Cf. A206545, and further crossrefs. given there.
%K A206546 nonn,easy,new
%O A206546 1,2
%A A206546 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 10 2012

%I A206553
%S A206553 5,5,13,7,5,5,7,7,11,13,13,11,5,11,37,11,5,23,13,47,89,13,19,19,11,7,
%T A206553 19,23,17,13,19,43,29,79,61,17,191,43,337,53,29,17,13,13,29,11,19,11,
%U A206553 11,13,43,163,29,13,7,53,23,97,31,29,41,83,79,23,191,97
%N A206553 Least prime p > 3 such that 2^n + p*2^floor((n+1)/2) - 1 is prime.
%H A206553 Pierre CAMI, <a href="/A206553/b206553.txt">Table of n, a(n) for n = 1..10000</a>
%e A206553 2^1+5*2^1-1 = 11 prime so a(1) = 5.
%e A206553 2^2+5*2^1-1 = 13 prime so a(2) = 5.
%o A206553 PFGW64 from Primeform group and SRYPTIFY
%o A206553 command : pfgw64 -f in.txt
%o A206553 in.txt file :
%o A206553 SCRIPT
%o A206553 DIM kk
%o A206553 DIM nn,0
%o A206553 DIM mm
%o A206553 DIMS tt
%o A206553 OPENFILEOUT myfil,prem.txt
%o A206553 LABEL loopn
%o A206553 SET nn,nn+1
%o A206553 IF nn>10000 THEN END
%o A206553 IF nn%2==0 THEN SET mm,nn/2
%o A206553 IF nn%2==1 THEN SET mm,nn/2+1
%o A206553 SET kk,2
%o A206553 LABEL loopk
%o A206553 SET kk,kk+1
%o A206553 SETS tt,%d,%d,%d,%d\ ;nn;kk;p(kk);mm
%o A206553 PRP (2^(nn-mm)+p(kk))*2^mm-1,tt
%o A206553 IF ISPRP THEN GOTO a
%o A206553 IF ISPRIME THEN GOTO a
%o A206553 GOTO loopk
%o A206553 LABEL a
%o A206553 WRITE myfil,tt
%o A206553 GOTO loopn
%o A206553 (Haskell)
%o A206553 a206553 n = head [p | p <- drop 2 a000040_list,
%o A206553                       a010051 (2^n + p*2^(div (n+1) 2) - 1) == 1]
%o A206553 -- Reinhard Zumkeller, Feb 10 2012
%Y A206553 Cf. A206554.
%Y A206553 Cf. A010051, A000040.
%K A206553 nonn,new
%O A206553 1,1
%A A206553 Pierre CAMI (pierre-cami(AT)bbox.fr), Feb 09 2012

%I A206554
%S A206554 1,4,5,8,9,12,16,17,33,61,92,113,137,216,237,833,872,948,1308,1728,
%T A206554 2509,4096,5457,6229,6432,6473,6868,7032,8128,8557,10449,11022,15629,
%U A206554 29217,40756,60684
%N A206554 Numbers n such that 2^n + 41*2^floor((n+1)/2) - 1 is prime.
%C A206554 All certified primes
%Y A206554 Cf. A206553.
%K A206554 nonn,new
%O A206554 1,2
%A A206554 Pierre CAMI (pierre-cami(AT)bbox.fr), Feb 09 2012

%I A206565
%S A206565 1,37,1368,50579,1870055,69141456,2556363817,94516319773,
%T A206565 3494547467784,129203739988235,4777043832096911,176621418047597472,
%U A206565 6530215423929009553,241441349267325755989,8926799707467123962040
%N A206565 Expansion of 1/(1-37*x+x^2).
%C A206565 Chebyshev polynomials S(n, 37).
%C A206565 A Diophantine property of these numbers: (a(n+1)-a(n-1))^2 - 1365*a(n)^2 = 4. - Bruno Berselli, Feb 09 2012
%H A206565 Bruno Berselli, <a href="/A206565/b206565.txt">Table of n, a(n) for n = 0..500</a>
%H A206565 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A206565 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A206565 <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (37,-1).
%F A206565 G.f.: 1/(1-37*x+x^2).
%F A206565 a(n) = Sum_{k, 0<=k<=n} A049310(n,k)*37^k.
%F A206565 a(n) = 37*a(n-1) - a(n-2), n>=1; a(0)=1, a(1) = 37, a(-1) = 0.
%F A206565 a(n) = -a(-n-2) = (t^(n+1)-1/t^(n+1))/(t-1/t) where t=(37+sqrt(1365))/2. - Bruno Berselli, Feb 09 2012
%F A206565 a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*36^k. - DELEHAM Philippe, Feb 10 2012
%o A206565 (PARI)  Vec(1/(1-37*x+x^2)+O(x^15))
%o A206565 (MAGMA)  Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-1365); S:=[(((37+r)/2)^n-1/((37+r)/2)^n)/r: n in [1..15]]; [Integers()!S[j]: j in [1..#S]];
%o A206565 (Maxima)  makelist(sum((-1)^k*binomial(n-k, k)*37^(n-2*k), k, 0, floor(n/2)), n, 0, 14);
%Y A206565 Cf. A144128, A078987.
%K A206565 nonn,easy,new
%O A206565 0,2
%A A206565 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 09 2012

%I A206591
%S A206591 1,1,2,3,28,485,5766,53767,430088,3459465,53303050,1746391691,
%T A206591 58977262092,1706810202253,42923448632078,965348202349455,
%U A206591 19877420584519696,385436337079476497,7654870637722391058,199927590326456092435,8556099311090244142100
%N A206591 E.g.f.: Sum_{n>=0} x^(n^2)*exp(n^2*x).
%C A206591 Compare to the partial theta series identity:
%C A206591 Sum_{n>=0} x^(n^2) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^(4*k-3))/(1-x^(4*k-1)).
%F A206591 E.g.f.: Sum_{n>=0} x^n*exp(n*x) * Product_{k=1..n} (1 - x^(4*k-3)*exp((4*k-3)*x))/(1 - x^(4*k-1)*exp((4*k-1)*x)).
%F A206591 Let q = x*exp(x), then the e.g.f. equals the continued fraction:
%F A206591 A(x) = 1/(1- q/(1- q*(q^2-1)/(1- q^5/(1- q^3*(q^4-1)/(1- q^9/(1- q^5*(q^6-1)/(1- q^13/(1- q^7*(q^8-1)/(1- ...))))))))), due to a partial elliptic theta function identity.
%e A206591 G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! + 28*x^4/4! + 485*x^5/5! +...
%e A206591 where the e.g.f. is defined by:
%e A206591 A(x) = 1 + x*exp(x) + x^4*exp(4*x) + x^9*exp(9*x) + x^16*exp(16*x) +...
%e A206591 Let q = x*exp(x), then the e.g.f. also equals the q-series:
%e A206591 A(x) = 1 + q*(1-q)/(1-q^3) + q^2*(1-q)*(1-q^5)/((1-q^3)*(1-q^7)) + q^3*(1-q)*(1-q^5)*(1-q^9)/((1-q^3)*(1-q^7)*(1-q^11)) +...
%o A206591 (PARI) {a(n)=n!*polcoeff(sum(m=0,sqrtint(n+1),x^(m^2)*exp(m^2*x+x*O(x^n))),n)}
%o A206591 (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(1+sum(m=1, n+1, x^m*exp(m*X)*prod(k=1, m,(1 - x^(4*k-3)*exp((4*k-3)*X))/(1 - x^(4*k-1)*exp((4*k-1)*X))) ), n)}
%o A206591 for(n=0, 35, print1(a(n), ", "))
%Y A206591 Cf. A193421, A206592.
%K A206591 nonn,new
%O A206591 0,3
%A A206591 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 09 2012

%I A206592
%S A206592 1,1,2,3,28,245,1446,6727,26888,459657,11208970,180639371,2158548492,
%T A206592 21024981133,176560640270,1324087390095,30001965127696,
%U A206592 1480628781891857,51566262458549778,1299527188916481811,25961751751545031700,436032724081792884501
%N A206592  E.g.f.: Sum_{n>=0} x^(n^2) * exp(n*x).
%C A206592  Compare to the partial theta series identity:
%C A206592 Sum_{n>=0} x^(n^2) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^(4*k-3))/(1-x^(4*k-1)).
%F A206592  E.g.f.: Sum_{n>=0} x^n*exp(n*x) * Product_{k=1..n} (1 - x^(4*k-3)*exp(x)) / (1 - x^(4*k-1)*exp(x)), due to a q-series identity.
%F A206592 The e.g.f. equals the continued fraction:
%F A206592 A(x) = 1/(1- x*exp(x)/(1- x*(x^2-1)*exp(x)/(1- x^5*exp(x)/(1- x^3*(x^4-1)*exp(x)/(1- x^9*exp(x)/(1- x^5*(x^6-1)*exp(x)/(1- x^13*exp(x)/(1- x^7*(x^8-1)*exp(x)/(1- ...))))))))), due to a partial elliptic theta function identity.
%e A206592  G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! + 28*x^4/4! + 245*x^5/5! +...
%e A206592 where the e.g.f. is defined by:
%e A206592 A(x) = 1 + x*exp(x) + x^4*exp(2*x) + x^9*exp(3*x) + x^16*exp(4*x) +...
%e A206592 By a q-series identity:
%e A206592 A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x^3*exp(x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x^5*exp(x))/((1-x^3*exp(x))*(1-x^7*exp(x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x^5*exp(x))*(1-x^9*exp(x))/((1-x^3*exp(x))*(1-x^7*exp(x))*(1-x^11*exp(x))) +...
%o A206592  (PARI) {a(n)=n!*polcoeff(sum(m=0,n+1,x^(m^2)*exp(m*x+x*O(x^n))),n)}
%o A206592 (PARI) {a(n)=local(A=1+x,X=x+x*O(x^n)); A=sum(m=0, n, x^m*exp(m*X)*prod(k=1, m, (1-x^(4*k-3)*exp(X))/(1-x^(4*k-1)*exp(X)))); n!*polcoeff(A, n)}
%o A206592  for(n=0, 35, print1(a(n), ", "))
%Y A206592  Cf. A193421, A206591.
%K A206592 nonn,new
%O A206592 0,3
%A A206592 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 09 2012

%I A206600
%S A206600 15,171,171,2532,9449,2532,38810,551631,551631,38810,597829,32258200,
%T A206600 122794254,32258200,597829,9217211,1886206112,27327609576,27327609576,
%U A206600 1886206112,9217211,142130380,110294918412,6080435385822,23138312968770
%N A206600 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206600 Table starts
%C A206600 .........15.............171.................2532...................38810
%C A206600 ........171............9449...............551631................32258200
%C A206600 .......2532..........551631............122794254.............27327609576
%C A206600 ......38810........32258200..........27327609576..........23138312968770
%C A206600 .....597829......1886206112........6080435385822.......19586354937923493
%C A206600 ....9217211....110294918412.....1352952771405899....16580357350118698482
%C A206600 ..142130380...6449441361133...301044368378558620.14035699363578442423160
%C A206600 .2191712741.377127604356885.66985084734298135608
%H A206600 R. H. Hardin, <a href="/A206600/b206600.txt">Table of n, a(n) for n = 1..59</a>
%e A206600 Some solutions for n=4 k=3
%e A206600 ..0..0..0..0....0..0..0..0....0..1..1..1....0..0..0..1....0..0..0..0
%e A206600 ..0..1..2..1....0..1..2..2....1..2..1..2....0..1..2..1....1..0..1..2
%e A206600 ..0..2..3..1....0..2..0..2....3..0..1..3....1..2..2..3....2..3..1..1
%e A206600 ..3..3..1..1....0..1..2..1....0..0..1..3....0..3..3..0....2..0..1..0
%e A206600 ..3..0..1..0....3..0..0..1....0..2..0..3....0..3..0..3....0..0..0..3
%K A206600 nonn,tabl,new
%O A206600 1,1
%A A206600 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206599
%S A206599 9217211,110294918412,1352952771405899,16580357350118698482,
%T A206599 203129458750387512198476
%N A206599 Number of (n+1)X7 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206599 Column 6 of A206600
%e A206599 Some solutions for n=4
%e A206599 ..0..0..0..1..0..1..0....0..1..1..0..0..2..3....0..0..0..1..0..1..0
%e A206599 ..0..1..2..2..0..1..0....3..0..0..2..1..3..2....0..1..2..1..1..0..1
%e A206599 ..2..0..1..1..1..2..3....0..1..0..2..0..1..2....1..3..2..3..2..0..2
%e A206599 ..2..2..3..1..3..3..0....1..3..0..1..1..3..2....3..0..1..0..1..2..2
%e A206599 ..0..3..1..3..0..2..0....2..0..2..2..1..1..0....0..3..2..3..0..1..0
%K A206599 nonn,new
%O A206599 1,1
%A A206599 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206598
%S A206598 597829,1886206112,6080435385822,19586354937923493,
%T A206598 63074271731085058880,203129458750387512198476
%N A206598 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206598 Column 5 of A206600
%e A206598 Some solutions for n=4
%e A206598 ..0..1..1..2..2..3....0..1..0..1..0..0....0..0..0..0..0..0....0..0..0..0..1..0
%e A206598 ..0..3..3..0..0..3....1..0..2..3..2..1....0..1..2..2..1..2....2..2..1..1..3..2
%e A206598 ..3..3..2..1..0..1....1..0..3..3..0..1....0..2..0..3..2..2....0..1..0..3..3..1
%e A206598 ..0..1..1..0..3..2....0..1..3..2..3..1....1..2..1..1..3..2....3..2..2..1..1..2
%e A206598 ..1..2..3..3..0..3....1..2..0..1..0..0....0..3..3..0..1..3....2..0..1..3..1..3
%K A206598 nonn,new
%O A206598 1,1
%A A206598 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206597
%S A206597 38810,32258200,27327609576,23138312968770,19586354937923493,
%T A206597 16580357350118698482,14035699363578442423160,
%U A206597 11881571779965774749880754,10058049575300573563980256082,8514392164624545092009786898646
%N A206597 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206597 Column 4 of A206600
%H A206597 R. H. Hardin, <a href="/A206597/b206597.txt">Table of n, a(n) for n = 1..59</a>
%e A206597 Some solutions for n=4
%e A206597 ..0..0..0..0..1....0..1..1..0..0....0..0..0..0..0....0..1..0..1..0
%e A206597 ..2..1..3..0..0....0..0..2..0..0....1..2..3..2..2....0..0..1..1..1
%e A206597 ..1..3..2..2..0....3..2..3..1..2....0..2..1..0..1....0..2..3..0..1
%e A206597 ..2..2..1..1..0....2..0..1..0..0....2..1..3..2..1....1..1..1..2..1
%e A206597 ..1..1..2..2..1....0..0..3..0..1....3..0..3..1..0....2..3..0..1..2
%K A206597 nonn,new
%O A206597 1,1
%A A206597 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206596
%S A206596 2532,551631,122794254,27327609576,6080435385822,1352952771405899,
%T A206596 301044368378558620,66985084734298135608,14904786179047249772283,
%U A206596 3316449502706465075744615,737939958180870807368858115
%N A206596 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206596 Column 3 of A206600
%H A206596 R. H. Hardin, <a href="/A206596/b206596.txt">Table of n, a(n) for n = 1..210</a>
%F A206596 Empirical: a(n) = 188*a(n-1) +5085*a(n-2) +538751*a(n-3) +7174168*a(n-4) +309429501*a(n-5) -1776082861*a(n-6) +28860632622*a(n-7) -359905728716*a(n-8) +2039075800421*a(n-9) -30459708500750*a(n-10) +174833793541111*a(n-11) -300293278143460*a(n-12) +4397179164685237*a(n-13) -34740353481829802*a(n-14) +252667207257063994*a(n-15) -1645737625781051525*a(n-16) +1494012905253706127*a(n-17) +7016560070067623526*a(n-18) +76250636913060634529*a(n-19) -91849233058660955961*a(n-20) -1313192270191739627435*a(n-21) +110708258209131272471*a(n-22) +6718782413596456239963*a(n-23) +17904054079780786238767*a(n-24) -19902748358486171917513*a(n-25) -90357037433300662770181*a(n-26) +46445505350553130768753*a(n-27) +2950156262169635403534*a(n-28) -351925328123940183340218*a(n-29) -1348773524441031306148778*a(n-30) +2320289764328887257556801*a(n-31) +5733881048375328976796963*a(n-32) +5705524487130220925381403*a(n-33) -4129763663354078121051878*a(n-34) -18824081542065485079480298*a(n-35) +6402061306263428361825365*a(n-36) -54648418465750083611143023*a(n-37) -34434863494730946433733616*a(n-38) +22875457596940403374259854*a(n-39) +168456136430463440566163050*a(n-40) +139329212119922460690166592*a(n-41) -210280867557016349286124992*a(n-42) -296620803307808911202487632*a(n-43) +197423562204775399833111400*a(n-44) +251880332423559645695653264*a(n-45) -22547688796924630026808928*a(n-46) -146149271363944797492462064*a(n-47) -94236527107917217827615488*a(n-48) +40465618821670955035582624*a(n-49) +31296678085789580806505728*a(n-50) +6191401187448994669092416*a(n-51) +24322993368343515722244352*a(n-52) -6447094900744642882702336*a(n-53) -12288897625302578809983488*a(n-54) +1920325887930781347131904*a(n-55) +736797463288573273427968*a(n-56) -133791108142820240271360*a(n-57) +65265035929730453483520*a(n-58) -11359380823472091340800*a(n-59) +3186497759104032768000*a(n-60) +102245622896443392000*a(n-61)
%e A206596 Some solutions for n=4
%e A206596 ..0..1..0..0....0..0..0..0....0..0..1..1....0..0..0..0....0..0..0..0
%e A206596 ..1..1..0..1....0..1..2..2....1..0..0..2....0..1..2..1....1..1..2..0
%e A206596 ..0..0..1..2....1..2..2..1....1..3..1..3....1..3..1..0....2..0..1..0
%e A206596 ..3..2..1..1....1..0..2..0....3..3..1..1....0..1..0..2....3..2..2..1
%e A206596 ..1..0..3..2....3..0..1..2....1..3..2..3....3..2..1..1....2..1..1..3
%K A206596 nonn,new
%O A206596 1,1
%A A206596 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206595
%S A206595 171,9449,551631,32258200,1886206112,110294918412,6449441361133,
%T A206595 377127604356885,22052335771550411,1289498594172126400,
%U A206595 75402743530876523488,4409135271445054483412,257821836876089274588349
%N A206595 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206595 Column 2 of A206600
%H A206595 R. H. Hardin, <a href="/A206595/b206595.txt">Table of n, a(n) for n = 1..210</a>
%F A206595 Empirical: a(n) = 51*a(n-1) +353*a(n-2) +4958*a(n-3) -2678*a(n-4) +14656*a(n-5) -160635*a(n-6) +19303*a(n-7) -859799*a(n-8) +242534*a(n-9) +14671*a(n-10) +3134369*a(n-11) +2183000*a(n-12) +2077689*a(n-13) -573423*a(n-14) -682544*a(n-15) -455304*a(n-16)
%e A206595 Some solutions for n=4
%e A206595 ..0..1..0....0..0..0....0..0..0....0..0..1....0..0..0....0..0..0....0..0..0
%e A206595 ..2..2..1....1..0..1....0..1..2....1..0..2....0..1..2....0..1..2....0..1..2
%e A206595 ..0..3..1....1..2..0....3..2..0....0..3..3....1..0..1....1..2..1....2..0..2
%e A206595 ..0..1..0....1..1..3....1..0..3....0..1..3....1..2..2....1..0..0....0..1..0
%e A206595 ..0..0..1....3..0..3....3..1..3....2..1..2....1..0..0....1..0..3....3..2..2
%K A206595 nonn,new
%O A206595 1,1
%A A206595 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206594
%S A206594 15,171,2532,38810,597829,9217211,142130380,2191712741,33797293408,
%T A206594 521171299118,8036724854275,123930361123673,1911068840711956,
%U A206594 29469647972560031,454436874891109270,7007646425074694210
%N A206594 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206594 Column 1 of A206600
%H A206594 R. H. Hardin, <a href="/A206594/b206594.txt">Table of n, a(n) for n = 1..210</a>
%F A206594 Empirical: a(n) = 16*a(n-1) -14*a(n-2) +98*a(n-3) -312*a(n-4) +114*a(n-5) -585*a(n-6) +72*a(n-7) -324*a(n-8) for n>9
%e A206594 Some solutions for n=4
%e A206594 ..0..1....0..1....0..0....0..0....0..0....0..0....0..1....0..0....0..0....0..1
%e A206594 ..2..0....1..2....1..0....0..1....1..1....1..0....0..2....1..2....0..0....0..0
%e A206594 ..2..2....1..2....2..3....0..2....1..0....0..2....1..1....1..3....1..2....1..1
%e A206594 ..1..0....3..0....2..1....3..2....2..3....0..3....1..0....0..0....2..3....0..0
%e A206594 ..3..2....1..2....1..0....3..1....0..1....0..2....3..2....1..3....2..3....1..0
%K A206594 nonn,new
%O A206594 1,1
%A A206594 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206593
%S A206593 15,9449,122794254,23138312968770,63074271731085058880
%N A206593 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than three equal edges, and new values 0..3 introduced in row major order
%C A206593 Diagonal of A206600
%e A206593 Some solutions for n=4
%e A206593 ..0..0..1..0..1....0..0..0..1..2....0..1..0..0..0....0..0..0..0..1
%e A206593 ..0..2..2..1..0....0..2..3..1..2....0..1..1..2..2....2..1..3..0..0
%e A206593 ..3..1..0..1..0....0..1..1..0..2....1..2..1..0..3....1..3..2..2..0
%e A206593 ..0..0..1..2..0....2..0..1..3..2....2..3..3..0..0....2..2..1..1..0
%e A206593 ..3..3..0..1..0....0..0..2..0..1....2..1..3..2..0....1..1..2..2..1
%K A206593 nonn,new
%O A206593 1,1
%A A206593 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A194705
%S A194705 7,4,3,2,2,3,1,1,3,2,0,1,1,2,3,1,0,1,1,2,2,0,1,0,1,1,2,2
%N A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (5 + m).
%C A194705 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 5. For further information see A182703 and A135010.
%F A194705 T(k,m) = A182703(5+m,k), with T(k,m) = 0 if k > 5+m.
%F A194705 T(k,m) = A194812(5+m,k).
%e A194705 Triangle begins:
%e A194705 7,
%e A194705 4, 3,
%e A194705 2, 2, 3,
%e A194705 1, 1, 3, 2,
%e A194705 0, 1, 1, 2, 3,
%e A194705 1, 0, 1, 1, 2, 2,
%e A194705 0, 1, 0, 1, 1, 2, 2,
%e A194705 ...
%e A194705 For k = 1 and  m = 1; T(1,1) = 7 because there are seven parts of size 1 in the outer shell of the partitions of 6, since 5 + m = 6, so a(1) = 7. For k = 2 and m = 1; T(2,1) = 4 because there are four parts of size 2 in the outer shell of the partitions of 6, since 5 + m = 6, so a(2) = 4.
%Y A194705 Always the sum of row k = p(5) = A000041(5) = 7.
%Y A194705 The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194704, this sequence, A194706-A194710.
%Y A194705 Cf. A135010, A138121, A194812.
%K A194705 nonn,tabl,more,new
%O A194705 1,1
%A A194705 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194704
%S A194704 5,1,4,1,2,2,0,1,1,3,1,0,1,1,2
%N A194704 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (4 + m).
%C A194704 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 4. For further information see A182703 and A135010.
%F A194704 T(k,m) = A182703(4+m,k), with T(k,m) = 0 if k > 4+m.
%F A194704 T(k,m) = A194812(4+m,k).
%e A194704 Triangle begins:
%e A194704 5,
%e A194704 1, 4,
%e A194704 1, 2, 2,
%e A194704 0, 1, 1, 3,
%e A194704 1, 0, 1, 1, 2,
%e A194704 ...
%e A194704 For k = 1 and  m = 1; T(1,1) = 5 because there are five parts of size 1 in the outer shell of the partitions of 5, since 4 + m = 5, so a(1) = 5. For k = 2 and m = 1; T(2,1) = 1 because there is only one part of size 2 in the outer shell of the partitions of 5, since 4 + m = 5, so a(2) = 1.
%Y A194704 Always the sum of row k = p(4) = A000041(4) = 5.
%Y A194704 The first (0-10) members of this family of triangles are A023531, A129186, A194702, A194703, this sequence, A194705-A194710.
%Y A194704 Cf. A135010, A138121, A182712-A182714, A194812.
%K A194704 nonn,tabl,more,new
%O A194704 1,1
%A A194704 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194703
%S A194703 3,2,1,0,1,2,1,0,1,1,0,1,0,1,1,0,0,1,0,1,1,3,2,1,0,1,2,1,0,1,1,0,1,0,
%T A194703 1,1,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,1,0,0,
%U A194703 0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0
%N A194703 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (3 + m).
%C A194703 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 3. For further information see A182703 and A135010.
%F A194703 T(k,m) = A182703(3+m,k), with T(k,m) = 0 if k > 3+m.
%F A194703 T(k,m) = A194812(3+m,k).
%e A194703 Triangle begins:
%e A194703 3,
%e A194703 2, 1,
%e A194703 0, 1, 2,
%e A194703 1, 0, 1, 1,
%e A194703 0, 1, 0, 1, 1,
%e A194703 0, 0, 1, 0, 1, 1,
%e A194703 0, 0, 0, 1, 0, 1, 1,
%e A194703 0, 0, 0, 0, 1, 0, 1, 1,
%e A194703 0, 0, 0, 0, 0, 1, 0, 1, 1,
%e A194703 0, 0, 0, 0, 0, 0, 1, 0, 1, 1,
%e A194703 ...
%e A194703 For k = 1 and  m = 1; T(1,1) = 3 because there are three parts of size 1 in the outer shell of the partitions of 4, since 3 + m = 4, so a(1) = 3. For k = 2 and m = 1; T(2,1) = 2 because there two parts of size 2 in the outer shell of the partitions of 4, since 3 + m = 4, so a(2) = 2.
%Y A194703 Always the sum of row k = p(3) = A000041(3) = 3.
%Y A194703 The first (0-10) members of this family of triangles are A023531, A129186, A194702, this sequence, A194704-A194710.
%Y A194703 Cf. A135010, A138121, A182712-A182714, A194812.
%K A194703 nonn,tabl,new
%O A194703 1,1
%A A194703 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A194702
%S A194702 2,0,2,1,0,1,0,1,0,1,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1,
%T A194702 0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,
%U A194702 0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1
%N A194702 Triangle read by rows: T(k,m) = number of occurrences of k in the outer shell of the partitions of (2 + m).
%C A194702 Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 2. For further information see A182703 and A135010.
%F A194702 T(k,m) = A182703(2+m,k), with T(k,m) = 0 if k > 2+m.
%F A194702 T(k,m) = A194812(2+m,k).
%e A194702 Triangle begins:
%e A194702 2,
%e A194702 0, 2,
%e A194702 1, 0, 1,
%e A194702 0, 1, 0, 1,
%e A194702 0, 0, 1, 0, 1,
%e A194702 0, 0, 0, 1, 0, 1,
%e A194702 0, 0, 0, 0, 1, 0, 1,
%e A194702 0, 0, 0, 0, 0, 1, 0, 1,
%e A194702 0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A194702 0, 0, 0, 0, 0, 0, 0, 1, 0, 1,
%e A194702 ...
%e A194702 For k = 1 and  m = 1; T(1,1) = 2 because there are two parts of size 1 in the outer shell of the partitions of 3, since 2 + m = 3, so a(1) = 2. For k = 2 and m = 1; T(2,1) = 0 because there are no parts of size 2 in the outer shell of the partitions of 3, since 2 + m = 3, so a(2) = 0.
%Y A194702 Always the sum of row k = p(2) = A000041(n) = 2.
%Y A194702 The first (0-10) members of this family of triangles are A023531, A129186, this sequence, A194703-A194710.
%Y A194702 Cf. A135010, A138121, A182712-A182714, A194812.
%K A194702 nonn,tabl,new
%O A194702 1,1
%A A194702 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A185313
%S A185313 2,2,5,5,5,17,17,53507,364187,155650237
%N A185313 Start of a sequence of n consecutive primes such that the sum of any three consecutive members is also prime.
%e A185313 Vacuously, the sum of every three consecutive members of {2, 3} is prime, so a(2) = 2.  a(4) = 5 because 5 + 7 + 11 and 7 + 11 + 13 are prime.
%o A185313 (PARI) a(n)=if(n<3,return(2),n-=2);my(len=0,p=2,q=3);forprime(r=5,default(primelimit),if(isprime(p+q+r),if(len++==n,my(t=p);for(i=2,n,t=precprime(t-1));return(t)),len=0);p=q;q=r) \\ Charles R Greathouse IV, Feb 08 2012
%K A185313 nonn,more,new
%O A185313 1,1
%A A185313 Charles R Greathouse IV (charles.greathouse(AT)case.edu), Feb 08 2012

%I A205598
%S A205598 0,1,10,11,101,110,111,1010,1011,1101,1110,1111,10110,10111,11010,
%T A205598 11011,11101,11110,11111,101011,101101,101110,101111,110110,110111,
%U A205598 111010,111011,111101,111110,111111,1011110,1011111
%N A205598 The number n written using a minimizing algorithm in the base where the values of the places are 1 and primes.
%C A205598 Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1 (See A007924 that uses a greedy algorithm for writing n). However in this sequence a(n) is generated by using a minimizing algorithm that gives the smallest binary vector for select members from the sequence Q = (1 union primes) that when summed gives n. Without the minimizing condition there is ambiguity - for example 8 = 7+1 = 5+3 = 5+2+1 has three representations.
%H A205598 Wikipedia, <a href="http://en.wikipedia.org/wiki/Complete_sequence"> Complete sequence</a>.
%F A205598 Let Q be the ordered sequence of (1 union primes), then a(n) x Q = n, where x is the inner product and the binary vector a(n) is in ascending powers of 2 with infinite trailing zeros.
%e A205598 8 = 7+1 = 5+3 = 5+2+1, so a(8) = 1011.
%t A205598 aprime[n_] := If[n==0, 1, Prime[n]]; seqtable[l_] := (stable=Table[aprime[j], {j, 0, l}]; stable); inttable[p_] := (itable=Reverse[IntegerDigits[p, 2]]; itable); h=1; otable={0}; ttable={}; While[h<100, (inttable[h]; seqtable[Length[itable]-1]; test=itable.stable; If[!MemberQ[ttable, test], AppendTo[otable, h], Null]; AppendTo[ttable, test]; h++)]; IntegerString[otable, 2]
%Y A205598 Cf. A007924, A185101, A200947
%K A205598 nonn,new
%O A205598 0,3
%A A205598 Frank M Jackson (fjackson(AT)matrix-logic.co.uk), Feb 08 2012

%I A193401
%S A193401 0,1,0,2,1,0,3,4,1,0,3,4,1,0,4,10,6,1,0,4,10,6,1,0,4,9,6,1,0,
%T A193401 4,9,6,1,0,5,20,21,8,1,0,5,20,21,8,1,0,5,20,21,8,1,0,5,18,20,
%U A193401 8,1,0,5,18,20,8,1,0,5,18,20,8,1,0,6,35,56,36,10,1,0,5,16,18,8,1,0,5,18,20,8,1
%V A193401 0,1,0,-2,1,0,3,-4,1,0,3,-4,1,0,-4,10,-6,1,0,-4,10,-6,1,0,-4,9,-6,1,0,
%W A193401 -4,9,-6,1,0,5,-20,21,-8,1,0,5,-20,21,-8,1,0,5,-20,21,-8,1,0,5,-18,20,
%X A193401 -8,1,0,5,-18,20,-8,1,0,5,-18,20,-8,1,0,-6,35,-56,36,-10,1,0,5,-16,18,-8,1,0,5,-18,20,-8,1
%N A193401 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable) of the characteristic polynomial of the Laplacian matrix of the rooted tree having Matula-Goedel number n.
%C A193401 Row n contains 1+A061775(n) entries (= 1+ number of vertices of the rooted tree).
%C A193401 The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%D A193401 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A193401 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A193401 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A193401 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A193401 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A193401 Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Goedel numbers 1..1000 (upper limit can be altered), then switches (easily) to the Laplacian matrices and finds the coefficients of their characteristic polynomials.
%e A193401 Row 4 is 0, 3, -4, 1 because the rooted tree having Matula-Goebel number 4 is V; the Laplacian matrix is [2,-1,-1; -1,1,0; -1,0,1], having characteristic polynomial x^3 - 4x^2 +3x
%e A193401 Triangle starts:
%e A193401 0, 1;
%e A193401 0, -2, 1;
%e A193401 0, 3, -4, 1;
%e A193401 0, 3, -4, 1;
%e A193401 0, -4, 10, -6, 1;
%e A193401 0, -4, 10, -6, 1;
%e A193401 0, -4, 9, -6, 1;
%e A193401 0, -4, 9, -6, 1;
%p A193401 with(numtheory): with(linalg): with(LinearAlgebra): DA := proc (d) local aa: aa := proc (i, j) if d[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(d), RowDimension(d), aa) end proc: AL := proc (a) local ll: ll := proc (i, j) if i = j then add(a[i, k], k = 1 .. RowDimension(a)) else -a[i, j] end if end proc: Matrix(RowDimension(a), RowDimension(a), ll) end proc: V := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else  end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 1000 do dd[n] := d(n) end do: for n from 1 to 18 do seq(coeff(CharacteristicPolynomial(AL(DA(d(n))), x), x, k), k = 0 .. V(n)) end do; # yields triangle in triangular form
%Y A193401 Cf. A061775, A184187, A192033
%K A193401 sign,tabf,new
%O A193401 1,4
%A A193401 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 09 2012

%I A201917
%S A201917 0,2085,2325,5253,6141,6293,7728,10013,11960,12920,14637,16940,17112,
%T A201917 18737,21648,21948,23541,24633,26588,27716,31620,33012,34937,35145,
%U A201917 38012,40641,42716,44268,47633,49848,52785,54237,56420,56840,60605,63828,67541,70448
%N A201917 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+84847)^2 = y^2.
%C A201917 Note that 84847 = 7 * 17 * 23 * 31, the first four primes in A058529.
%F A201917 a(n) = a(n-1) + 6*a(n-81) - 6*a(n-82) - a(n-162) + a(n-163), where the 163 initial terms can be computed using the Mathematica program. The initial terms begin with 0 and end with 1696940.
%t A201917 d = 84847; t = Select[Range[0,68000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]
%Y A201917 Cf. A201916 (has list of all such sequences).
%K A201917 nonn,new
%O A201917 1,2
%A A201917 T. D. Noe (noe(AT)sspectra.com), Feb 09 2012

%I A201916
%S A201916 0,75,203,323,552,708,1020,1127,1311,1428,1608,1820,1955,2336,2675,
%T A201916 3128,3311,3627,3927,4140,4508,4743,5535,6003,6800,7280,7848,8211,
%U A201916 8588,9240,9860,11063,11895,13583,14168,15180,15827,16827,18011,18768,20915,22836
%N A201916 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+2737)^2 = y^2.
%C A201916 Note that 2737 = 7 * 17 * 23, the product of the first three distinct primes in A058529 (and A001132) and hence the smallest such number. This sequence satisfies a linear difference equation of order 55 whose 55 initial terms can be found by running the Mathematica program.
%C A201916 There are many sequences like this one. What determines the order of the linear difference equation? All primes p have order 7. For those p, it appears that p^2 has order 11, p^3 order 15, and p^i order 3+4*i. It appears that for semiprimes p*q (with p > q), the order is 19. What is the next term of the sequence beginning 3, 7, 19, 55, 163? This could be sequence A052919, which is 1 + 2*3^f, where f is the number of primes.
%C A201916 The crossref list is thought to be complete up to today.
%F A201916 a(n) = a(n-1) + 6*a(n-27) - 6*a(n-28) - a(n-54) + a(n-55), where the 55 initial terms can be computed using the Mathematica program.
%t A201916 d = 2737; terms = 100; t = Select[Range[0, 55000], IntegerQ[Sqrt[#^2 + (#+d)^2]] &]; Do[AppendTo[t, t[[-1]] + 6*t[[-27]] - 6*t[[-28]] - t[[-54]] + t[[-55]]], {terms-55}]; t
%Y A201916 Cf. A001652 (1), A076296 (7), A118120 (17), A118337 (23), A118674 (31).
%Y A201916 Cf. A129288 (41), A118675 (47), A118554 (49), A118673 (71), A129289 (73).
%Y A201916 Cf. A118676 (79), A129298 (89), A129836 (97), A157119 (103), A161478 (113).
%Y A201916 Cf. A129837 (119), A129992 (127), A129544 (137), A161482 (151).
%Y A201916 Cf. A206426 (161), A130608 (167), A161486 (191), A185394 (193).
%Y A201916 Cf. A129993 (199), A198294 (217), A130609 (223), A129625 (233).
%Y A201916 Cf. A204765 (239), A129991 (241), A129626 (281), A205644 (287).
%Y A201916 Cf. A129640 (313), A205672 (329), A129999 (337), A118611 (343).
%Y A201916 Cf. A130610 (359), A129641 (409), A130645 (439), A130004 (449).
%Y A201916 Cf. A129642 (457), A129725 (521), A101152 (569), A130005 (577).
%Y A201916 Cf. A111258 (601), A115135 (617), A130013 (647), A130646 (727).
%Y A201916 Cf. A122694 (761), A123654 (809), A129010 (833), A130647 (839).
%Y A201916 Cf. A129857 (857), A130014 (881), A129974 (937), A129975 (953).
%Y A201916 Cf. A130017 (967), A118630 (2401), A118576 (16807), A201917 (84847).
%K A201916 nonn,new
%O A201916 1,2
%A A201916 T. D. Noe (noe(AT)sspectra.com), Feb 09 2012

%I A206577
%S A206577 4,16,16,61,256,61,232,3721,3721,232,882,53824,188699,53824,882,3353,
%T A206577 777924,9509750,9509750,777924,3353,12747,11242609,478456896,
%U A206577 1665431164,478456896,11242609,12747,48460,162486009,24071410239,290967246344
%N A206577 T(n,k)=Number of nXk 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206577 Table starts
%C A206577 .....4.........16.............61.................232.....................882
%C A206577 ....16........256...........3721...............53824..................777924
%C A206577 ....61.......3721.........188699.............9509750...............478456896
%C A206577 ...232......53824........9509750..........1665431164............290967246344
%C A206577 ...882.....777924......478456896........290967246344.........176372042982658
%C A206577 ..3353...11242609....24071410239......50835598921675......106921561045262424
%C A206577 .12747..162486009..1211164182464....8882668617521828....64829265966214645776
%C A206577 .48460.2348371600.60940227961282.1552079697677787063.39306166321588974667092
%H A206577 R. H. Hardin, <a href="/A206577/b206577.txt">Table of n, a(n) for n = 1..84</a>
%e A206577 Some solutions for n=4 k=3
%e A206577 ..2..2..1....2..1..2....0..0..3....3..1..1....0..3..3....1..1..0....0..0..3
%e A206577 ..0..0..3....3..0..1....0..1..1....1..3..3....0..1..3....3..2..3....2..3..1
%e A206577 ..0..0..2....2..0..0....2..0..2....3..1..3....2..1..1....3..0..0....1..2..0
%e A206577 ..2..2..0....1..3..0....2..1..0....1..3..3....2..1..1....2..1..3....1..0..0
%K A206577 nonn,tabl,new
%O A206577 1,1
%A A206577 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206576
%S A206576 12747,162486009,1211164182464,8882668617521828,64829265966214645776,
%T A206576 473233846073429226663536
%N A206576 Number of nX7 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206576 Column 7 of A206577
%e A206576 Some solutions for n=4
%e A206576 ..0..3..3..0..0..3..3....1..3..2..0..0..0..0....1..2..3..2..3..1..1
%e A206576 ..3..2..1..0..1..0..1....1..0..1..3..3..3..2....1..3..3..3..1..3..1
%e A206576 ..1..0..3..2..1..2..3....3..0..3..0..2..0..2....3..3..3..1..1..1..3
%e A206576 ..3..0..3..2..3..0..1....2..3..0..0..0..3..2....2..3..0..1..1..1..1
%K A206576 nonn,new
%O A206576 1,1
%A A206576 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206575
%S A206575 3353,11242609,24071410239,50835598921675,106921561045262424,
%T A206575 224922074723556223280,473233846073429226663536,
%U A206575 995612236558478493078872933,2094486828658187985171032792568
%N A206575 Number of nX6 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206575 Column 6 of A206577
%H A206575 R. H. Hardin, <a href="/A206575/b206575.txt">Table of n, a(n) for n = 1..11</a>
%e A206575 Some solutions for n=4
%e A206575 ..0..0..2..2..0..3....3..2..2..0..2..1....2..0..1..0..2..0....1..0..2..3..2..1
%e A206575 ..3..0..1..0..1..2....0..1..2..1..3..2....1..2..1..3..0..0....1..0..3..3..0..1
%e A206575 ..1..2..0..0..2..0....1..3..0..0..2..0....1..3..0..2..0..2....0..0..3..2..3..1
%e A206575 ..3..2..1..2..0..0....2..1..0..3..3..3....3..1..1..0..2..1....1..2..0..1..0..0
%K A206575 nonn,new
%O A206575 1,1
%A A206575 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206574
%S A206574 882,777924,478456896,290967246344,176372042982658,106921561045262424,
%T A206574 64829265966214645776,39306166321588974667092,
%U A206574 23830450089557555202846497,14447747775590219278502903910,8759230508354654974219180831199
%N A206574 Number of nX5 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206574 Column 5 of A206577
%H A206574 R. H. Hardin, <a href="/A206574/b206574.txt">Table of n, a(n) for n = 1..14</a>
%e A206574 Some solutions for n=4
%e A206574 ..3..2..3..0..1....0..0..2..1..1....2..2..0..1..2....0..0..2..0..0
%e A206574 ..1..1..1..3..1....0..2..1..0..1....3..2..0..1..2....1..1..3..3..0
%e A206574 ..2..3..0..1..3....0..2..2..1..0....1..1..0..0..2....3..1..1..3..2
%e A206574 ..0..0..2..3..0....3..1..0..1..0....1..3..3..1..2....1..1..1..0..3
%K A206574 nonn,new
%O A206574 1,1
%A A206574 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206573
%S A206573 232,53824,9509750,1665431164,290967246344,50835598921675,
%T A206573 8882668617521828,1552079697677787063,271191336278876503161,
%U A206573 47384416611699348366558,8279310662890040257782573
%N A206573 Number of nX4 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206573 Column 4 of A206577
%H A206573 R. H. Hardin, <a href="/A206573/b206573.txt">Table of n, a(n) for n = 1..210</a>
%e A206573 Some solutions for n=4
%e A206573 ..0..1..3..0....1..3..0..1....1..0..2..3....3..1..1..1....1..0..3..2
%e A206573 ..2..0..3..0....0..0..2..0....2..1..1..0....3..3..3..1....1..1..0..0
%e A206573 ..2..0..3..2....0..1..1..3....3..3..0..1....3..1..3..3....3..1..3..0
%e A206573 ..1..2..0..2....3..0..3..1....0..0..3..2....3..1..1..1....1..0..1..3
%K A206573 nonn,new
%O A206573 1,1
%A A206573 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206572
%S A206572 61,3721,188699,9509750,478456896,24071410239,1211164182464,
%T A206572 60940227961282,3066196250895643,154274761365491709,
%U A206572 7762277323583045839,390556037506212530634,19650677663071356937032,988716329728016793029948
%N A206572 Number of nX3 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206572 Column 3 of A206577
%H A206572 R. H. Hardin, <a href="/A206572/b206572.txt">Table of n, a(n) for n = 1..210</a>
%e A206572 Some solutions for n=4
%e A206572 ..3..2..2....1..3..2....2..0..1....1..0..0....3..3..2....0..0..3....0..0..3
%e A206572 ..0..2..3....2..0..2....0..2..0....0..0..3....0..0..2....2..3..1....0..1..1
%e A206572 ..3..0..1....2..1..1....1..2..1....1..2..1....1..1..0....1..2..0....2..0..2
%e A206572 ..2..3..2....0..1..1....3..0..0....0..1..3....1..0..0....1..0..0....2..1..0
%K A206572 nonn,new
%O A206572 1,1
%A A206572 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206571
%S A206571 16,256,3721,53824,777924,11242609,162486009,2348371600,33940324441,
%T A206571 490529342884,7089470711236,102461949277801,1480851163185241,
%U A206571 21402288176353681,309320714104227225,4470517516089946896
%N A206571 Number of nX2 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206571 Column 2 of A206577
%H A206571 R. H. Hardin, <a href="/A206571/b206571.txt">Table of n, a(n) for n = 1..210</a>
%F A206571 Empirical: a(n) = 16*a(n-1) -12*a(n-2) -166*a(n-3) +134*a(n-4) +1595*a(n-5) -1291*a(n-6) -11538*a(n-7) +9401*a(n-8) +344*a(n-9) -2098*a(n-10) +2832*a(n-11) -1207*a(n-12) -846*a(n-13) +548*a(n-14) +637*a(n-15) -488*a(n-16) +102*a(n-17) -13*a(n-18) -161*a(n-19) +85*a(n-20) +20*a(n-21) -14*a(n-22) -4*a(n-23) +6*a(n-24) -4*a(n-25) +a(n-26) +2*a(n-27) -a(n-28)
%e A206571 Some solutions for n=4
%e A206571 ..2..2....1..3....1..0....2..0....0..0....0..2....3..1....2..3....3..3....1..0
%e A206571 ..1..0....0..1....2..3....1..0....3..0....1..1....1..2....3..0....0..1....1..3
%e A206571 ..0..3....0..0....2..1....2..0....1..2....3..1....0..1....1..2....0..0....1..3
%e A206571 ..0..0....2..0....1..0....1..2....2..0....0..0....0..0....3..2....3..2....3..1
%K A206571 nonn,new
%O A206571 1,1
%A A206571 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206570
%S A206570 4,16,61,232,882,3353,12747,48460,184229,700378,2662606,10122349,
%T A206570 38481829,146295209,556166085,2114359836,8038098037,30558194945,
%U A206570 116172168341,441648229594,1679000758004,6383006556982,24266083569200
%N A206570 Number of nX1 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206570 Column 1 of A206577
%H A206570 R. H. Hardin, <a href="/A206570/b206570.txt">Table of n, a(n) for n = 1..210</a>
%F A206570 Empirical: a(n) = 4*a(n-1) -3*a(n-3) +2*a(n-5) -a(n-7)
%e A206570 Some solutions for n=4
%e A206570 ..1....1....2....0....0....1....0....3....0....1....1....0....1....2....1....1
%e A206570 ..0....3....0....2....1....3....2....0....0....0....1....3....3....0....0....3
%e A206570 ..0....2....1....2....1....0....1....3....2....1....0....1....3....1....1....2
%e A206570 ..0....1....2....1....3....1....3....1....0....3....3....2....1....0....1....2
%K A206570 nonn,new
%O A206570 1,1
%A A206570 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206569
%S A206569 4,256,188699,1665431164,176372042982658,224922074723556223280
%N A206569 Number of nXn 0..3 arrays avoiding the pattern z-1 z-1 z in any row, column or nw-to-se diagonal
%C A206569 Diagonal of A206577
%e A206569 Some solutions for n=4
%e A206569 ..0..2..2..1....1..1..1..3....0..1..0..3....3..1..1..3....1..0..2..3
%e A206569 ..1..0..1..1....3..3..2..0....2..2..1..2....0..1..0..0....2..1..1..0
%e A206569 ..2..2..2..0....0..2..1..1....1..1..3..3....0..0..3..0....3..3..0..1
%e A206569 ..0..3..0..0....3..1..0..0....0..3..3..0....2..1..2..0....0..0..3..2
%K A206569 nonn,new
%O A206569 1,1
%A A206569 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206239
%S A206239 0,1,4,24,309,8807,530356,65679652,16512273616,8370804628178,
%T A206239 8525389679187197,17408681737224080093,71192609533782031405771,
%U A206239 582711051458083440858730497,9542765396943645975520145941300,312620974584432225019935558843189172,20485270547000003746699189223065768145956
%N A206239 Subsequence A005318(t_n), where t_n is the n-th triangula number (A000217).
%C A206239 These are the repeated terms in A205744.
%H A206239  R. K. Guy, <a href="http://dx.doi.org/10.1016/S0304-0208(08)73500-X">Sets of integers whose subsets have distinct sums</a>, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982. See Table 1.
%Y A206239 Cf. A005318, A205744, A096858.
%K A206239 nonn,new
%O A206239 1,3
%A A206239 N. J. A. Sloane (njas(AT)research.att.com), Feb 09 2012

%I A205744
%S A205744 0,0,1,1,2,4,4,7,13,24,24,44,84,161,309,309,594,1164,2284,4484,8807,
%T A205744 8807,17305,34301,68008,134852,267420,530356,530356,1051905,2095003,
%U A205744 4172701,8311101,16554194,32973536,65679652,65679652,130828948,261127540,521203175,1040311347,2076449993
%N A205744 The sequence "u_{n-r}" used by Conway and Guy in the construction of A005318 and A096858.
%H A205744 R. K. Guy, <a href="http://dx.doi.org/10.1016/S0304-0208(08)73500-X">Sets of integers whose subsets have distinct sums</a>, pp. 141-154 of Theory and practice of combinatorics. Ed. A. Rosa, G. Sabidussi and J. Turgeon. Annals of Discrete Mathematics, 12. North-Holland 1982. See Table 1, column (3).
%F A205744 This is A005318 with the terms A005318(i) repeated iff i is a triangular number.
%Y A205744 Cf. A005318, A096858.
%K A205744 nonn,new
%O A205744 1,5
%A A205744 N. J. A. Sloane (njas(AT)research.att.com), Feb 09 2012

%I A206543
%S A206543 1,3,5,7,9,9,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1,3,5,7,
%T A206543 9,9,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1,3,5,7,9,9,7,5,3,1,1
%N A206543 Period 10: repeat 1, 3, 5, 7, 9, 9, 7, 5, 3, 1.
%C A206543 For general Mod n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Mod 11 for the positive odd numbers not divisible by 11, which are given in A204454.
%C A206543 The underlying period length 22 sequence with offset 0 is P_11, also called Mod11, periodic([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]).
%F A206543 a(n) = A204454(n) (Mod 11) := Mod11(A204454(n)), with the periodic sequence Mod11 with period length 22 given in the comment section.
%F A206543 O.g.f.: x*(1+x^9+3*x*(1+x^7)+5*x^2*(1+x^5)+7*x^3*(1+x^3)+9*x^4*(1+x))/(1-x^10) = x*(1+x)*(1-x^5)/((1+x^5)*(1-x)^2).
%e A206543 Residue Mod 11 of the positive odd numbers not divisible by 11:
%e A206543 A204454: 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, ...
%e A206543 Mod 11:  1, 3, 5, 7, 9,  9,  7,  5,  3,  1,  1,  3,  5, ...
%Y A206543 A000012 (Mod 3), A084101 (Mod 5), A110551 (Mod 7).
%K A206543 nonn,easy,new
%O A206543 1,2
%A A206543 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 09 2012

%I A206545
%S A206545 1,3,5,7,9,11,13,15,15,13,11,9,7,5,3,1,1,3,5,7,9,11,13,15,15,13,11,9,
%T A206545 7,5,3,1,1,3,5,7,9,11,13,15,15,13,11,9,7,5,3,1,1,3,5,7,9,11,13,15,15,
%U A206545 13,11,9,7,5,3,1,1,3,5,7,9,11,13,15,15,13,11,9,7,5,3,1
%N A206545 Period length 16: repeat 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1.
%C A206545 For general Mod n see a comment on A203571. This sequence gives the Mod 17 residues of the odd numbers not divisible by 17, which are given in A204458.
%C A206545 The underlying periodic sequence with period length 34 is
%C A206545 periodic([0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,0,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1]). This sequence with offset 0 is called P_17 or Mod17.
%F A206545 a(n) = A204458(n) (Mod 17) := Mod17(A204458(n)), n>=1, with the periodic sequence Mod17, with period length 34, defined in the comment section.
%F A206545 O.g.f.: x*(1+x^15+3*x*(1+x^13)+5*x^2*(1+x^11)+7*x^3*(1+x^9)+9*x^4*(1+x^7)+11*x^5*(1+x^5)+ 13*x^6*(1+x^3)+15*x^7*(1+x))/(1-x^16) = x*(1+x)^2*(1+x^2)*(1+x^4)/((1+x^8)*(1-x)).
%e A206545 Residue Mod 17 of the positive odd numbers not divisible by 17:
%e A206545 A204458: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35,...
%e A206545 Mod 17:  1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11,  9,  7,  5,  3,  1,  1,...
%Y A206545 Cf. A000012 (Mod 3), A084101 (Mod 5), A110551 (Mod 7), A206543 (Mod 11), A206544 (Mod 13).
%K A206545 nonn,easy,new
%O A206545 1,2
%A A206545 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 09 2012

%I A206544
%S A206544 1,3,5,7,9,11,11,9,7,5,3,1,1,3,5,7,9,11,11,9,7,5,3,1,1,3,5,7,9,11,11,
%T A206544 9,7,5,3,1,1,3,5,7,9,11,11,9,7,5,3,1,1,3,5,7,9,11,11,9,7,5,3,1,1,3,5,
%U A206544 7,9,11,11,9,7,5,3,1
%N A206544 Period 12: repeat 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1.
%C A206544 For general Mod n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Mod 13 of the positive odd numbers not divisible by 13, which are given in A204457.
%C A206544 The underlying periodic sequence with period length 26 is periodic([0,1,2,3,4,5,6,7,8,9,10,11,12,0,12,11,10,9,8,7,6,5,4,3,2,1]), called, with offset 0, P_13 or Mod13.
%F A206544 a(n) = A204457(n) (Mod 13) := Mod13(A204457(n)), n>=1, with the period length 26 periodic sequence Mod13 given in the comment section.
%F A206544 O.g.f.: x*(1+x^11+3*x*(1+x^9)+5*x^2*(1+x^7)+7*x^3*(1+x^5)+9*x^4*(1+x^3)+11*x^5*(1+x))/(1-x^12) = x*(1-x^6)*(1+x)/((1+x^6)*(1-x)^2).
%e A206544 Residue Mod 13 of the positive odd numbers not divisible by 13:
%e A206544 A204457: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, ...
%e A206544 Mod 13:  1, 3, 5, 7, 9, 11, 11,  9,  7,  5,  3,  1,  1,  3,  5,  7, ...
%Y A206544 Cf. A000012 (Mod 3), A084101 (Mod 5), A110551 (Mod 7), A206543 (Mod 11).
%K A206544 nonn,easy,new
%O A206544 1,2
%A A206544 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 09 2012

%I A205672
%S A205672 0,87,112,184,235,376,451,595,660,987,1440,1575,1971,2256,3055,3484,
%T A205672 4312,4687,6580,9211,9996,12300,13959,18612,21111,25935,28120,39151,
%U A205672 54484,59059,72487,82156,109275,123840,151956,164691,228984,318351,345016,423280
%N A205672 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+329)^2 = y^2.
%t A205672 LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,87,112,184,235,376,451,595,660,987,1440,1575,1971,2256,3055,3484,4312,4687,6580}, 120]
%Y A205672 Cf. A185394, A198294, A203619, A204765, A205644, A206426
%K A205672 nonn,new
%O A205672 1,2
%A A205672 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 09 2012

%I A205644
%S A205644 0,25,36,205,252,273,328,705,748,861,988,1045,1968,2233,2352,2665,
%T A205644 4836,5085,5740,6477,6808,12177,13720,14413,16236,28885,30336,34153,
%U A205644 38448,40377,71668,80661,84700,95325,169048,177505,199752,224785,236028,418405,470820
%N A205644 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+287)^2 = y^2.
%t A205644 LinearRecurrence[ {1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,25,36,205,252,273,328,705,748,861,988,1045,1968,2233,2352,2665,4836,5085,5740}, 70]
%Y A205644 Cf. A185394, A198294, A203619, A204765, A206426
%K A205644 nonn,new
%O A205644 1,2
%A A205644 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 09 2012

%I A204765
%S A204765 0,217,220,717,1900,1917,4780,11661,11760,28441,68544,69121,166344,
%T A204765 400081,403444,970101,2332420,2352021,5654740,13594917,13709160,
%U A204765 32958817,79237560,79903417,192098640,461830921,465711820,1119633501,2691748444,2714367981
%N A204765 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+239)^2 = y^2.
%t A204765 LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,217,220,717,1900,1917,4780}, 70]
%Y A204765 Cf. A185394, A198294, A203619, A206426
%K A204765 nonn,new
%O A204765 1,2
%A A204765 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 09 2012

%I A206417
%S A206417 0,3,7,14,25,43,72,119,195,318,517,839,1360,2203,3567,5774,9345,15123,
%T A206417 24472,39599,64075,103678,167757,271439,439200,710643,1149847,1860494,
%U A206417 3010345,4870843,7881192,12752039,20633235,33385278,54018517,87403799,141422320
%N A206417 Lucas-Fibonacci numbers of the form (5F(n)+3L(n)-8)/2.
%D A206417 Thomas Koshy, Fibonacci and Lucas Numbers and Applications. Wiley, New York, (2001): PAGE NUMBER I WILL HOPEFULLY GET TOMORROW
%H A206417 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (2,0,-1).
%t A206417 Table[(5*Fibonacci[n] + 3*LucasL[n] - 8)/2, {n, 60}] (* or *) Table[LucasL[n+2] - 4, {n, 60}]
%o A206417 (PARI) a(n)=fibonacci(n)+3*fibonacci(n+1)-4 \\ Charles R Greathouse IV, Feb 08 2012
%Y A206417 Cf. A000032, A000045.
%K A206417 nonn,easy,new
%O A206417 1,2
%A A206417 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A198294
%S A198294 0,63,68,135,155,248,276,407,420,651,980,1007,1376,1488,2015,2175,
%T A198294 2928,3003,4340,6251,6408,8555,9207,12276,13208,17595,18032,25823,
%U A198294 36960,37875,50388,54188,72075,77507,103076,105623,151032,215943,221276,294207,316355
%N A198294 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+217)^2 = y^2.
%t A198294 LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,63,68,135,155,248,276,407,420,651,980,1007,1376,1488,2015,2175,2928,3003,4340}, 110]
%Y A198294 Cf. A185394, A206426
%K A198294 nonn,new
%O A198294 1,2
%A A198294 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 09 2012

%I A185394
%S A185394 0,152,203,579,1403,1692,3860,8652,10335,22967,50895,60704,134328,
%T A185394 297104,354275,783387,1732115,2065332,4566380,10095972,12038103,
%U A185394 26615279,58844103,70163672,155125680,342969032,408944315,904139187,1998970475,2383502604,5269709828
%N A185394 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+193)^2 = y^2.
%t A185394 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,152,203,579,1403,1692,3860},70]
%Y A185394 Cf. A206426
%K A185394 nonn,new
%O A185394 1,2
%A A185394 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 09 2012

%I A185391
%S A185391 0,1,10,114,1556,25080,468462,9971920,238551336,6339784320,
%T A185391 185391061010,5917263922944,204735466350780,7633925334590464,
%U A185391 305188474579874550,13023103577435351040,590850477768105474128,28401410966866912051200,1441935117039649859464986
%N A185391 Sum_{k =0...n} A185390(n,k) * k.
%C A185391 The total number of elements, x in the domain of definition of all partial functions on n labelled objects such that for all i in {1,2,3,...} (f^i)(x) is defined.
%t A185391 nn=20; tx=Sum[n^(n-1) x^n/n!,{n,1,nn}]; txy=Sum[n^(n-1) (x y)^n/n!, {n,1,nn}]; f[list_] := Select[list, #>0&];
%t A185391   D[Range[0,nn]! CoefficientList[Series[Exp[tx]/(1-txy),{x,0,nn}],x],y]/.y->1
%Y A185391 Cf. A076728.
%K A185391 nonn,new
%O A185391 0,3
%A A185391 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 09 2012

%I A185390
%S A185390 1,1,1,3,2,4,16,9,12,27,125,64,72,108,256,1296,625,640,810,1280,3125,
%T A185390 16807,7776,7500,8640,11520,18750,46656,262144,117649,108864,118125,
%U A185390 143360,196875,326592,823543
%N A185390 Triangular array read by rows. T(n,k) is the number of partial functions on n labelled objects in which the domain of definition contains exactly k elements such that for all i in {1,2,3,...}, (f^i)(x) is defined.
%C A185390 Here, for any x in the domain of definition (f^i)(x) denotes the i-fold composition of f with itself, e.g., (f^2)(x)=f(f(x)).  Domain of definition is the set of all values x for which f(x) is defined.
%C A185390 T(n,n)=n^n  The partial functions that are total functions.
%C A185390 T(n,0)=A000272(offset) See comment and link by Dennis Walsh.
%D A185390 Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132, II.21.
%F A185390 E.g.f.: exp(T(x))/(1-T(x*y)) where T(x) is the e.g.f. for A000169.
%e A185390   1
%e A185390   1     1
%e A185390   3     2     4
%e A185390   16    9     12    27
%e A185390   125   64    72    108   256
%e A185390   1296  625   640   810   1280   3125
%e A185390   16807 7776  7500  8640  11520  18750  46656
%t A185390 nn = 7; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; txy = Sum[n^(n - 1) (x y)^n/n!, {n, 1, nn}]; f[list_] := Select[list, # > 0 &]; Map[f, Range[0, nn]! CoefficientList[Series[Exp[tx]/(1 - txy), {x, 0, nn}], {x, y}]] // Flatten
%K A185390 nonn,new
%O A185390 0,4
%A A185390 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 09 2012

%I A206300
%S A206300 1,2,6,32,210,1536,12012,98304,831402,7208960,63740820,572522496,
%T A206300 5209363380,47915728896,444799488600,4161823309824,39209074920090,
%U A206300 371626340253696,3541117629057540
%V A206300 -1,2,6,32,210,1536,12012,98304,831402,7208960,63740820,572522496,
%W A206300 5209363380,47915728896,444799488600,4161823309824,39209074920090,
%X A206300 371626340253696,3541117629057540
%N A206300 Expand the real root of y^3 - y + x in powers of x, then multiply coefficient of x^n by -4^n to get integers.
%C A206300 This equation has degree three, whereas the Catalan numbers (A000108) arise from an equation of degree two as developed in the Andrews reference.
%C A206300 FindSequenceFunction[] gives:
%C A206300 f[n_] = If[n == 2, 2, (
%C A206300   3 2^(-3 + 2 n) (-4 + (3 n)/2)!)/((-2 + n/2)! (n - 1)!)]
%C A206300 Table[f[n], {n, 1, 31}]
%C A206300 The binomial version is:
%C A206300 f[n_] = If[n == 1, -1,
%C A206300   If[n == 2, 2, (
%C A206300    3 2^(-3 + 2 n) Binomial[3*n/2 - 4, n/2 - 2])/(n - 1)]]
%C A206300 Table[f[n], {n, 1, 31}]
%C A206300 Order four also exists which makes me think this is a general sequence of sequences:
%C A206300 p[x_] = y /. Solve[y^4 - y + x == 0, y][[4]]
%C A206300 b = Table[-36^n*
%C A206300    FullSimplify[
%C A206300     ExpandAll[SeriesCoefficient[ Series[p[x], {x, 0, 30}], n]
%C A206300      ]], {n, 0, 30}]
%D A206300 G. E. Andrews, Number Theory, 1971, Dover Publications New York,
%D A206300 pp. 41 - 43
%t A206300 p[x_] = y /. Solve[y^3 - y + x == 0, y][[1]]
%t A206300 b = Table[-4^n*FullSimplify[ExpandAll[SeriesCoefficient[ Series[p[x], {x, 0, 30}], n]]], {n, 0, 30}]
%Y A206300 Cf. A000108
%K A206300 sign,easy,new
%O A206300 0,2
%A A206300 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 05 2012
%E A206300 Edited by N. J. A. Sloane, Feb 09 2012

%I A184187
%S A184187 0,1,1,0,1,4,6,0,1,4,6,0,1,12,32,20,0,1,12,32,20,0,1,12,
%T A184187 28,15,0,1,12,28,15,0,1,32,120,140,50,0,1,32,120,140,50,
%U A184187 0,1,32,120,140,50,0,1,32,112,116,38,0,1,32,112,116,38,0,1,32,112,116,38,0,1
%V A184187 0,1,-1,0,1,-4,-6,0,1,-4,-6,0,1,-12,-32,-20,0,1,-12,-32,-20,0,1,-12,
%W A184187 -28,-15,0,1,-12,-28,-15,0,1,-32,-120,-140,-50,0,1,-32,-120,-140,-50,
%X A184187 0,1,-32,-120,-140,-50,0,1,-32,-112,-116,-38,0,1,-32,-112,-116,-38,0,1,-32,-112,-116,-38,0,1
%N A184187 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable) of the characteristic polynomial of the distance matrix of the rooted tree having Matula-Goedel number n.
%C A184187 Row n contains 1+A061775(n) entries (= 1+ number of vertices of the rooted tree). The pairs 0,1 are ends of rows.
%C A184187 The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%D A184187 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A184187 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A184187 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A184187 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A184187 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A184187 Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Goedel numbers 1..1000 (upper limit can be altered) and then finds the coefficients of their characteristic polynomials.
%e A184187 Row 4 is -4,-6,0,1 because the rooted tree having Matula-Goebel number 4 is V; the distance matrix is [0,1,1; 1,0,2; 1,2,0], having characteristic polynomial -4-6x+x^3.
%e A184187 Triangle starts:
%e A184187 0,1;
%e A184187 -1,0,1;
%e A184187 -4,-6,0,1;
%e A184187 -4,-6,0,1;
%e A184187 -12,-32,-20,0,1;
%e A184187 -12,-32,-20,0,1;
%e A184187 -12,-28,-15,0,1;
%p A184187 with(numtheory): with(linalg): with(LinearAlgebra): V := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else  end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 1000 do dd[n] := d(n) end do: for n to 14 do seq(coeff(CharacteristicPolynomial(d(n), x), x, k), k = 0 .. V(n)) end do;
%Y A184187 Cf. A061775
%K A184187 sign,tabf,new
%O A184187 1,6
%A A184187 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2012

%I A200521
%S A200521 420,630,660,780,840,924,990,1020,1050,1092,1140,1170,1260,1320,1380,
%T A200521 1386,1428,1470,1530,1540,1560,1596,1638,1650,1680,1710,1716,1740,
%U A200521 1820,1848,1860,1890,1932,1950,1980,2040,2070,2100,2142,2184,2220,2244
%N A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.
%C A200521 I expect that A123709(a(k))=32.
%H A200521 M. F. Hasler, <a href="/A200521/b200521.txt">Table of n, a(n) for n = 1..18347</a>
%o A200521 (PARI) is_A200521(n,c=4)={ omega(n)==c & bigomega(n)>c }
%Y A200521 Cf. A200511, A178212.
%K A200521 nonn,new
%O A200521 1,1
%A A200521 M. F. Hasler (oeis2012-removeThis(AT)hasler.fr), Feb 09 2012

%I A185352
%S A185352 2,4,8,17,39,92,275,922,2894
%N A185352 The "smallest countdown" numbers are the smallest positive integer that can not be made using the numbers n through 1, in order, using the operations +,-,*,/, and parentheses.
%C A185352 Inspired by a now-lost blog post where someone discussed a "new year's countdown" equation for 2012, e.g. 10 * (9 + ((8 * (((7 + (6 / (5 * 4))) * 3) + 2)) + 1)) = 2012.  This sequence has been "verified" by two independently created programs.
%e A185352 for n = 3, a(3) = 8, because 3*2+1=7, and 3*(2+1)=9, but there is no equation with 3,2,and 1 in order that equals 8. Note that if we allow the order to change, we can make 8, because 2*(3+1)=8, but reordering is not allowed.
%o A185352 # Python
%o A185352 from fractions import Fraction
%o A185352 def genAllTrees(l):
%o A185352 ....if len(l) == 0:
%o A185352 ........return
%o A185352 ....elif len(l) == 1:
%o A185352 ........yield l[0], str(l[0])
%o A185352 ....else:
%o A185352 ........for middle in range(len(l)):
%o A185352 ............for lval, leqn in genAllTrees(l[:middle]):
%o A185352 ................for rval, reqn in genAllTrees(l[middle:]):
%o A185352 ....................yield lval+rval, ("(" + leqn + " + " + reqn + ")")
%o A185352 ....................yield lval-rval, ("(" + leqn + " - " + reqn + ")")
%o A185352 ....................yield lval*rval, ("(" + leqn + " * " + reqn + ")")
%o A185352 ....................if rval != Fraction(0):
%o A185352 ........................yield lval/rval, ("(" + leqn + " / " + reqn + ")")
%o A185352 .
%o A185352 def findSmallestIntNotPresent(n):
%o A185352 ....vals = {}
%o A185352 ....for val, eqn in genAllTrees([Fraction(i) for i in range(n, 0, -1)]):
%o A185352 ........if val.denominator == 1:
%o A185352 ............val = val.numerator
%o A185352 ............if val not in vals:
%o A185352 ................vals[val] = eqn
%o A185352 ....i = 1
%o A185352 ....while i in vals:
%o A185352 ........i += 1
%o A185352 ....return i
%o A185352 .
%o A185352 for i in range(1,11):
%o A185352 ....print(i, findSmallestIntNotPresent(i))
%Y A185352 Related to A060315, which is the smallest number that can not be made with the numbers 1 to n, in any order.
%K A185352 nonn,hard,more,new
%O A185352 1,1
%A A185352 Peter Boothe (peter.boothe(AT)manhattan.edu), Feb 08 2012 Abraham Asfaw (aasfaw.student(AT)manhattan.edu), Feb 08 2012

%I A192033
%S A192033 0,1,1,0,1,0,2,0,1,0,2,0,1,1,0,3,0,1,1,0,3,0,1,0,0,3,0,1,0,0,3,
%T A192033 0,1,0,3,0,4,0,1,0,3,0,4,0,1,0,3,0,4,0,1,0,2,0,4,0,1,0,2,0,4,0,1,
%U A192033 0,2,0,4,0,1,1,0,6,0,5,0,1,0,0,0,4,0,1,0,2,0,4,0,1,1,0,5,0,5,0,1
%V A192033 0,1,-1,0,1,0,-2,0,1,0,-2,0,1,1,0,-3,0,1,1,0,-3,0,1,0,0,-3,0,1,0,0,-3,
%W A192033 0,1,0,3,0,-4,0,1,0,3,0,-4,0,1,0,3,0,-4,0,1,0,2,0,-4,0,1,0,2,0,-4,0,1,
%X A192033 0,2,0,-4,0,1,-1,0,6,0,-5,0,1,0,0,0,-4,0,1,0,2,0,-4,0,1,-1,0,5,0,-5,0,1
%N A192033 Triangle read by rows: row n contains the coefficients (of the increasing powers of the variable) of the characteristic polynomial of the adjacency matrix of the rooted tree having Matula-Goedel number n.
%C A192033 Row n contains 1+A061775(n) entries (= 1+ number of vertices of the rooted tree). The pairs 0,1 are the ends of rows.
%C A192033 The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%D A192033 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A192033 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A192033 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A192033 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A192033 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A192033 Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Goedel numbers 1..1000 (upper limit can be altered), then switches (easily) to adjacency matrices and finds the coefficients of their characteristic polynomials.
%e A192033 Row 4 is 0,-2, 0, 1 because the rooted tree having Matula-Goebel number 4 is V; the adjacency matrix is [0,1,1; 1,0,0; 1,0,0], having characteristic polynomial x^3 - 2x.
%e A192033 Triangle starts:
%e A192033 0, 1;
%e A192033 -1, 0, 1;
%e A192033 0, -2, 0, 1;
%e A192033 0,-2, 0, 1;
%e A192033 1, 0, -3, 0, 1;
%e A192033 1, 0, -3, 0, 1;
%e A192033 0, 0, -3, 0, 1;
%p A192033 with(numtheory): with(linalg): with(LinearAlgebra): DA := proc (d) local aa: aa := proc (i, j) if d[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(d), RowDimension(d), aa) end proc: V := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else  end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 1000 do dd[n] := d(n) end do: for n to 18 do seq(coeff(CharacteristicPolynomial(DA(d(n)), x), x, k), k = 0 .. V(n)) end do; # yields sequence in triangular form
%Y A192033 A061775, A184187
%K A192033 sign,tabf,new
%O A192033 1,7
%A A192033 Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 08 2012

%I A200511
%S A200511 12,18,20,24,28,36,40,44,45,48,50,52,54,56,63,68,72,75,76,80,88,92,96,
%T A200511 98,99,100,104,108,112,116,117,124,135,136,144,147,148,152,153,160,
%U A200511 162,164,171,172,175,176,184,188,189,192,196,200,207,208,212,216,224,225,232,236
%N A200511 Numbers n with bigomega(n)>2 but omega(n)=2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.
%C A200511 It appears that this is equal to A123711.
%o A200511 (PARI) for(n=1,999,bigomega(n)>2 & omega(n)==2 & print1(n","))
%K A200511 nonn,new
%O A200511 1,1
%A A200511 M. F. Hasler (oeis2012-removeThis(AT)hasler.fr), Feb 09 2012

%I A206542
%S A206542 81,399,399,1620,2577,1620,6462,14781,14781,6462,26481,91674,141759,
%T A206542 91674,26481,110769,594348,1419921,1419921,594348,110769,469617,
%U A206542 3902463,14528199,22647363,14528199,3902463,469617,2009553,25900263,150387285
%N A206542 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206542 Table starts
%C A206542 ......81.......399........1620..........6462...........26481............110769
%C A206542 .....399......2577.......14781.........91674..........594348...........3902463
%C A206542 ....1620.....14781......141759.......1419921........14528199.........150387285
%C A206542 ....6462.....91674.....1419921......22647363.......369441411........6062752884
%C A206542 ...26481....594348....14528199.....369441411......9567269487......249853070682
%C A206542 ..110769...3902463...150387285....6062752884....249853070682....10370489275269
%C A206542 ..469617..25900263..1567749888..100259781690...6556642024200...433165769694465
%C A206542 .2009553.172835859.16417626507.1661323349889.172503406693200.18115074102295746
%H A206542 R. H. Hardin, <a href="/A206542/b206542.txt">Table of n, a(n) for n = 1..179</a>
%e A206542 Some solutions for n=4 k=3
%e A206542 ..0..1..0..0....2..2..0..0....1..1..2..1....0..0..2..2....1..0..2..0
%e A206542 ..0..0..0..2....2..0..0..2....1..2..2..2....0..2..2..0....2..1..0..1
%e A206542 ..0..1..0..0....2..0..2..2....1..1..2..0....2..2..0..0....0..2..1..0
%e A206542 ..0..0..0..1....0..0..2..1....2..1..1..1....2..0..0..1....1..0..2..1
%e A206542 ..1..0..2..2....2..2..2..1....2..1..0..2....2..2..2..1....2..1..0..2
%K A206542 nonn,tabl,new
%O A206542 1,1
%A A206542 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206541
%S A206541 469617,25900263,1567749888,100259781690,6556642024200,
%T A206541 433165769694465,28770371143057311,1915604623565617890,
%U A206541 127722868372348334343,8521820836493432131926,568800839142318486776736
%N A206541 Number of (n+1)X8 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206541 Column 7 of A206542
%H A206541 R. H. Hardin, <a href="/A206541/b206541.txt">Table of n, a(n) for n = 1..93</a>
%e A206541 Some solutions for n=4
%e A206541 ..0..1..2..1..2..0..1..2....2..1..0..2..0..2..1..2....2..2..0..1..1..1..0..0
%e A206541 ..1..2..0..2..1..2..0..1....0..2..1..0..1..0..2..1....2..0..0..0..1..0..0..1
%e A206541 ..0..1..2..0..2..0..1..2....2..0..2..1..0..1..0..2....2..2..2..0..1..0..1..1
%e A206541 ..2..0..1..2..0..1..2..1....1..2..1..2..1..2..1..0....1..1..2..0..0..0..0..0
%e A206541 ..0..1..2..0..2..0..1..0....2..0..2..0..2..0..2..1....1..2..2..0..2..2..2..1
%K A206541 nonn,new
%O A206541 1,1
%A A206541 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206540
%S A206540 110769,3902463,150387285,6062752884,249853070682,10370489275269,
%T A206540 433165769694465,18115074102295746,759196260042210954,
%U A206540 31822407182233848810,1334925952358094973038,55997558706291917520756
%N A206540 Number of (n+1)X7 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206540 Column 6 of A206542
%H A206540 R. H. Hardin, <a href="/A206540/b206540.txt">Table of n, a(n) for n = 1..210</a>
%e A206540 Some solutions for n=4
%e A206540 ..1..2..2..2..1..0..2....2..1..0..0..0..2..1....2..0..2..2..1..0..0
%e A206540 ..1..1..2..1..1..0..2....2..1..0..2..2..2..1....0..0..0..2..1..0..1
%e A206540 ..1..2..2..1..0..0..2....1..1..0..0..2..1..1....2..2..0..2..1..1..1
%e A206540 ..1..1..2..1..0..2..2....1..0..0..2..2..2..1....0..2..0..2..1..0..1
%e A206540 ..2..2..2..1..0..2..0....0..0..2..2..1..1..1....2..2..0..2..1..1..1
%K A206540 nonn,new
%O A206540 1,1
%A A206540 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206539
%S A206539 26481,594348,14528199,369441411,9567269487,249853070682,
%T A206539 6556642024200,172503406693200,4545389492802288,119878356782179572,
%U A206539 3163274433238575696,83498862564488470557,2204480938367057407965,58208728563598806225396
%N A206539 Number of (n+1)X6 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206539 Column 5 of A206542
%H A206539 R. H. Hardin, <a href="/A206539/b206539.txt">Table of n, a(n) for n = 1..210</a>
%e A206539 Some solutions for n=4
%e A206539 ..1..1..2..1..1..0....2..1..0..1..1..2....2..1..2..1..0..0....0..2..1..2..1..2
%e A206539 ..1..2..2..1..0..0....1..1..0..0..1..1....2..1..1..1..0..2....2..1..2..0..2..0
%e A206539 ..2..2..1..1..0..2....0..0..0..1..1..0....2..1..2..1..0..2....0..2..0..1..0..1
%e A206539 ..1..1..1..0..0..2....2..2..0..1..0..0....1..1..2..1..0..2....1..0..1..2..1..0
%e A206539 ..0..0..0..0..2..2....1..2..0..1..0..2....0..1..1..1..0..0....2..1..0..1..2..1
%K A206539 nonn,new
%O A206539 1,1
%A A206539 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206538
%S A206538 6462,91674,1419921,22647363,369441411,6062752884,100259781690,
%T A206538 1661323349889,27605998634595,459099135099705,7643138171689107,
%U A206538 127288176601858407,2120715004190021418,35337966665490442968
%N A206538 Number of (n+1)X5 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206538 Column 4 of A206542
%H A206538 R. H. Hardin, <a href="/A206538/b206538.txt">Table of n, a(n) for n = 1..210</a>
%F A206538 Empirical: a(n) = 48*a(n-1) -511*a(n-2) -10370*a(n-3) +248056*a(n-4) -15316*a(n-5) -39319672*a(n-6) +229682760*a(n-7) +3101791467*a(n-8) -34508533564*a(n-9) -110666023609*a(n-10) +2737134599902*a(n-11) -1944425563980*a(n-12) -138867498519160*a(n-13) +454259584497554*a(n-14) +4717128238889588*a(n-15) -27608187524865471*a(n-16) -102274180218808104*a(n-17) +1043996333252379617*a(n-18) +943477829221661450*a(n-19) -28016705891644261101*a(n-20) +23261406089897667612*a(n-21) +556274293200286508015*a(n-22) -1247499078646449613354*a(n-23) -8197520625284603552352*a(n-24) +30956284714277574520008*a(n-25) +85788210608452981488138*a(n-26) -530512013424753881671808*a(n-27) -516952150443611363033893*a(n-28) +6860471761383109750519032*a(n-29) -1196032896481459102230229*a(n-30) -69021658631054450401808738*a(n-31) +74690607572116965833811745*a(n-32) +543183882272307214778119452*a(n-33) -1068832023445491188455812561*a(n-34) -3282566901670169285954654670*a(n-35) +10121075007155708620346009855*a(n-36) +14208182659535945876339618952*a(n-37) -72428793981671823713894486303*a(n-38) -32121231858726786079218334254*a(n-39) +408561833169761505869290715091*a(n-40) -97308349221930835302587736596*a(n-41) -1841328547308946011319804982233*a(n-42) +1496086757780849194670707779590*a(n-43) +6607513220724960922143316804746*a(n-44) -9213989262423698472453477802160*a(n-45) -18434504153086579417527704082968*a(n-46) +39442327691765676067799592211840*a(n-47) +37228177774441591640934398952255*a(n-48) -129534694907220060624836181675896*a(n-49) -40411622888314093343038080005821*a(n-50) +336731593320828848352682392100038*a(n-51) -48000975598652004135942114782295*a(n-52) -697940645406503285491347495692428*a(n-53) +366061991293902005485294586186825*a(n-54) +1143881348647212863635157279370810*a(n-55) -1036179391820524735349062785273390*a(n-56) -1441003912455915547339751581754552*a(n-57) +1994307396560010385145001622894484*a(n-58) +1293057280966202915929507409775364*a(n-59) -2880450359808654988899259730426769*a(n-60) -610274996777128366435342681486352*a(n-61) +3206204387719753625082069162148931*a(n-62) -309572560120321475994259000367938*a(n-63) -2750925460606335489132885799897064*a(n-64) +966596217324838111166926215121460*a(n-65) +1784942039300871554423500119764452*a(n-66) -1080019864206003990836334339795984*a(n-67) -834157860762937714117084886744187*a(n-68) +782078830446037238134570376553724*a(n-69) +244721848194879595446885644836151*a(n-70) -400312061002262348775412251599866*a(n-71) -17194469109658921524060267687464*a(n-72) +146280505270289981074600431916512*a(n-73) -22008582387456425838893453087556*a(n-74) -37124185790721383988932283290328*a(n-75) +11956439527550103074989437626432*a(n-76) +6042730178489268592604840524544*a(n-77) -3255912945245123127193382837184*a(n-78) -479699417345292019690689572736*a(n-79) +535241841461358252995313057856*a(n-80) -20333215259394138725609269120*a(n-81) -52468101920892556966516786688*a(n-82) +8870143959963568197241974784*a(n-83) +2677763029393438955566882816*a(n-84) -861537094407627597994369024*a(n-85) -32179558459945322846375936*a(n-86) +37442235660396113187921920*a(n-87) -2725899345794501790760960*a(n-88) -649016325155956378959872*a(n-89) +101959108007770572980224*a(n-90) +933451572057030000640*a(n-91) -944251229793536704512*a(n-92) +47259023090814812160*a(n-93) for n>96
%e A206538 Some solutions for n=4
%e A206538 ..1..2..1..1..1....1..0..0..2..2....1..2..2..0..1....0..2..1..2..2
%e A206538 ..1..1..1..2..2....1..0..2..2..0....1..2..0..0..0....2..2..1..1..1
%e A206538 ..1..2..2..2..0....0..0..0..2..2....1..2..0..1..0....0..2..2..1..0
%e A206538 ..1..1..1..1..1....1..0..2..2..0....1..2..0..0..0....0..2..1..1..1
%e A206538 ..1..2..1..0..2....1..0..0..2..2....2..2..0..1..0....2..2..1..2..1
%K A206538 nonn,new
%O A206538 1,1
%A A206538 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206537
%S A206537 1620,14781,141759,1419921,14528199,150387285,1567749888,16417626507,
%T A206537 172416155157,1814112395499,19110547682607,201479426103099,
%U A206537 2125267836114957,22425737682248088,236688761385595083
%N A206537 Number of (n+1)X4 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206537 Column 3 of A206542
%H A206537 R. H. Hardin, <a href="/A206537/b206537.txt">Table of n, a(n) for n = 1..210</a>
%F A206537 Empirical: a(n) = 26*a(n-1) -169*a(n-2) -962*a(n-3) +15074*a(n-4) -23548*a(n-5) -388835*a(n-6) +1661306*a(n-7) +3387121*a(n-8) -32428954*a(n-9) +18217134*a(n-10) +298256916*a(n-11) -598690104*a(n-12) -1238898284*a(n-13) +4947375888*a(n-14) +248363732*a(n-15) -19612727472*a(n-16) +17680860320*a(n-17) +37546981780*a(n-18) -68886915424*a(n-19) -19805672744*a(n-20) +117856544624*a(n-21) -46294103104*a(n-22) -94823075520*a(n-23) +86888550208*a(n-24) +22051127568*a(n-25) -55468199168*a(n-26) +13452820320*a(n-27) +12246999328*a(n-28) -7344098112*a(n-29) +380505984*a(n-30) +639121152*a(n-31) -167178240*a(n-32) +11612160*a(n-33) for n>37
%e A206537 Some solutions for n=4
%e A206537 ..2..1..1..1....0..1..0..0....2..1..0..2....0..0..2..2....2..2..0..0
%e A206537 ..1..1..2..1....0..0..0..2....1..2..1..0....0..2..2..0....2..0..0..2
%e A206537 ..1..2..2..2....0..1..0..0....2..1..0..1....2..2..0..0....2..0..2..2
%e A206537 ..1..2..0..2....0..0..0..1....0..2..1..0....2..0..0..1....0..0..2..1
%e A206537 ..1..2..2..2....1..0..2..2....1..0..2..1....2..2..2..1....2..2..2..1
%K A206537 nonn,new
%O A206537 1,1
%A A206537 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206536
%S A206536 399,2577,14781,91674,594348,3902463,25900263,172835859,1158343314,
%T A206536 7783336557,52398118053,353178544149,2382554812068,16081864446159,
%U A206536 108592014091443,733452273626199,4954756282052310,33475276596812193
%N A206536 Number of (n+1)X3 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206536 Column 2 of A206542
%H A206536 R. H. Hardin, <a href="/A206536/b206536.txt">Table of n, a(n) for n = 1..210</a>
%F A206536 Empirical: a(n) = 14*a(n-1) -49*a(n-2) -96*a(n-3) +814*a(n-4) -820*a(n-5) -2734*a(n-6) +5904*a(n-7) +127*a(n-8) -8794*a(n-9) +6847*a(n-10) -336*a(n-11) -1164*a(n-12) +288*a(n-13) for n>17
%e A206536 Some solutions for n=4
%e A206536 ..0..2..2....2..1..2....2..0..0....1..1..0....2..1..0....1..2..2....0..1..0
%e A206536 ..0..1..2....2..1..2....0..0..2....0..0..0....2..1..0....2..2..1....1..2..1
%e A206536 ..0..1..1....1..1..1....0..2..2....0..1..1....2..1..1....0..2..2....2..1..0
%e A206536 ..1..1..2....0..0..0....2..2..0....1..1..0....1..1..0....0..2..0....0..2..1
%e A206536 ..1..2..2....1..1..1....0..0..0....1..0..0....0..0..0....2..2..2....1..0..2
%K A206536 nonn,new
%O A206536 1,1
%A A206536 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206535
%S A206535 81,399,1620,6462,26481,110769,469617,2009553,8655441,37451697,
%T A206535 162572721,707284113,3081856785,13442980209,58681197681,256284824145,
%U A206535 1119691708881,4893034223409,21386003380017,93482466850833,408662174944401
%N A206535 Number of (n+1)X2 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206535 Column 1 of A206542
%H A206535 R. H. Hardin, <a href="/A206535/b206535.txt">Table of n, a(n) for n = 1..210</a>
%F A206535 Empirical: a(n) = 7*a(n-1) -9*a(n-2) -15*a(n-3) +18*a(n-4) for n>8
%e A206535 Some solutions for n=4
%e A206535 ..1..2....1..1....0..0....0..2....2..2....2..0....2..0....0..0....1..1....2..1
%e A206535 ..1..2....0..0....1..1....1..1....2..0....2..2....2..2....0..2....1..1....0..2
%e A206535 ..1..1....2..0....1..0....1..2....0..0....2..0....1..2....0..2....1..1....1..0
%e A206535 ..2..1....0..0....0..0....0..0....1..0....2..2....2..2....2..2....1..1....2..1
%e A206535 ..1..1....0..1....2..2....2..1....0..0....0..2....2..0....1..0....1..1....1..0
%K A206535 nonn,new
%O A206535 1,1
%A A206535 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206534
%S A206534 81,2577,141759,22647363,9567269487,10370489275269,28770371143057311,
%T A206534 203028344528664369537,3648979640901069855030843
%N A206534 Number of (n+1)X(n+1) 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing, and anticlockwise edge increases nonincreasing, rightwards and downwards
%C A206534 Diagonal of A206542
%e A206534 Some solutions for n=4
%e A206534 ..1..2..1..2..1....0..2..1..2..2....1..2..0..2..1....0..1..0..2..0
%e A206534 ..1..1..1..1..1....2..2..1..1..1....0..1..2..0..2....1..2..1..0..1
%e A206534 ..2..1..2..1..2....0..2..2..1..0....2..0..1..2..0....0..1..2..1..0
%e A206534 ..1..1..1..1..1....0..2..1..1..1....0..1..2..0..2....1..2..0..2..1
%e A206534 ..0..0..0..0..0....2..2..1..2..1....2..0..1..2..1....0..1..2..0..2
%K A206534 nonn,new
%O A206534 1,1
%A A206534 R. H. Hardin (rhhardin(AT)att.net) Feb 09 2012

%I A206524
%S A206524 6,14,16,36,50,53,54,56,57,63,65,74,77,78,81,84,86,88,95,96,101,102,
%T A206524 107,109,113,115,116,127,132,134,136,137,141,142,148,150,151,154,155,
%U A206524 163,166,168,173,177,180,181,182,185,188,192,196,197,200,207,209,213,216,218,221,222,223,224,225,226,229,231,234,237,239,240,241,243,244,247,254
%N A206524 By definition, the sequences A004210, A206522 and A206523 are disjoint; the present sequence gives the complement of their union.
%Y A206524 Cf. A004210, A206522, A206524.
%K A206524 nonn,new
%O A206524 1,1
%A A206524 N. J. A. Sloane (njas(AT)research.att.com), Feb 08 2012

%I A206523
%S A206523 4,9,11,19,21,26,31,33,38,44,46,48,51,61,68,70,73,75,85,91,93,97,98,
%T A206523 108,110,120,123,125,130,133,140,152,157,162,164,165,169,179,189,191,
%U A206523 203,204,205,210,212,220,228,232,245,251,261,263,268,269,278,283,290,292,303,306,308,313,323,324,327,335,348,350,363,372,382,389,391,395,396,406,417,418,419,421
%N A206523 The pairwise sums of the terms of A004210.
%Y A206523 Cf. A004210, A206522, A206524.
%K A206523 nonn,new
%O A206523 1,1
%A A206523 N. J. A. Sloane (njas(AT)research.att.com), Feb 08 2012

%I A206521
%S A206521 15,186,186,2786,10998,2786,43804,695541,695541,43804,697369,44350086,
%T A206521 177097738,44350086,697369,11136544,2830665213,45142367398,
%U A206521 45142367398,2830665213,11136544,177977851,180689239504,11507555684346
%N A206521 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206521 Table starts
%C A206521 .........15.............186..................2786......................43804
%C A206521 ........186...........10998................695541...................44350086
%C A206521 .......2786..........695541.............177097738................45142367398
%C A206521 ......43804........44350086...........45142367398.............45954984167446
%C A206521 .....697369......2830665213........11507555684346..........46782397564239329
%C A206521 ...11136544....180689239504......2933481908580803.......47624714678427763425
%C A206521 ..177977851..11534052523389....747797195466172046....48482197805424282951252
%C A206521 .2844864002.736261766040550.190626930556386514119.49355119922405928499997192
%H A206521 R. H. Hardin, <a href="/A206521/b206521.txt">Table of n, a(n) for n = 1..143</a>
%e A206521 Some solutions for n=4 k=3
%e A206521 ..0..0..1..2....0..0..0..0....0..1..1..0....0..0..0..0....0..0..0..0
%e A206521 ..0..3..3..0....0..1..1..1....0..0..2..0....0..0..1..1....0..0..1..1
%e A206521 ..1..0..1..2....0..0..0..1....0..2..1..1....1..2..0..2....1..0..1..1
%e A206521 ..1..1..0..0....0..1..0..0....3..1..0..0....0..1..0..3....2..2..1..0
%e A206521 ..1..0..3..2....0..1..1..1....1..0..2..3....2..2..1..2....0..3..0..0
%K A206521 nonn,tabl,new
%O A206521 1,1
%A A206521 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206520
%S A206520 177977851,11534052523389,747797195466172046,48482197805424282951252,
%T A206520 3143262221646752596264102690,203788158413867986023089008758269,
%U A206520 13212265000747034021229857133763537610
%N A206520 Number of (n+1)X8 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206520 Column 7 of A206521
%H A206520 R. H. Hardin, <a href="/A206520/b206520.txt">Table of n, a(n) for n = 1..31</a>
%e A206520 Some solutions for n=4
%e A206520 ..0..0..0..0..0..1..1..1....0..1..1..0..0..0..2..0....0..0..0..0..0..0..0..0
%e A206520 ..0..2..2..2..3..1..2..0....0..2..1..1..3..1..0..0....0..1..0..0..2..0..0..1
%e A206520 ..1..0..2..0..3..3..0..3....1..0..2..3..2..1..1..0....0..3..2..3..3..3..1..1
%e A206520 ..3..0..2..1..3..0..1..1....3..3..0..1..0..0..3..2....0..3..1..0..1..2..0..2
%e A206520 ..0..1..1..1..3..2..0..0....3..1..1..2..0..1..0..0....1..0..1..1..3..1..2..0
%K A206520 nonn,new
%O A206520 1,1
%A A206520 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206519
%S A206519 11136544,180689239504,2933481908580803,47624714678427763425,
%T A206519 773181099472666620872437,12552495871839218224290213767,
%U A206519 203788158413867986023089008758269,3308474580174074486151948611032236902
%N A206519 Number of (n+1)X7 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206519 Column 6 of A206521
%H A206519 R. H. Hardin, <a href="/A206519/b206519.txt">Table of n, a(n) for n = 1..86</a>
%e A206519 Some solutions for n=4
%e A206519 ..0..0..1..0..0..2..0....0..0..1..0..0..0..2....0..1..0..1..1..1..0
%e A206519 ..3..0..2..0..3..1..1....3..1..1..2..0..1..0....2..0..0..2..3..3..3
%e A206519 ..2..0..2..0..1..0..3....0..0..0..3..0..1..1....2..1..0..0..3..2..1
%e A206519 ..2..2..1..2..1..1..3....2..0..2..2..1..0..3....2..3..3..3..3..0..1
%e A206519 ..2..0..3..3..0..1..3....2..2..0..2..3..3..0....1..1..1..3..3..3..3
%K A206519 nonn,new
%O A206519 1,1
%A A206519 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206518
%S A206518 697369,2830665213,11507555684346,46782397564239329,
%T A206518 190187445308482279868,773181099472666620872437,
%U A206518 3143262221646752596264102690,12778503510703479899595167222949
%N A206518 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206518 Column 5 of A206521
%H A206518 R. H. Hardin, <a href="/A206518/b206518.txt">Table of n, a(n) for n = 1..210</a>
%e A206518 Some solutions for n=4
%e A206518 ..0..1..1..0..2..3....0..1..0..0..0..0....0..1..0..1..0..0....0..0..1..0..2..0
%e A206518 ..2..0..2..3..0..1....0..0..2..1..3..1....1..0..2..3..2..1....0..0..0..1..0..1
%e A206518 ..0..1..2..2..0..3....3..0..0..1..0..2....1..0..3..3..0..1....3..3..3..2..3..0
%e A206518 ..2..3..0..1..0..2....0..1..0..2..0..1....0..0..3..2..3..1....3..0..2..0..2..2
%e A206518 ..1..2..2..3..1..0....1..1..1..3..2..2....1..2..0..1..0..0....3..0..0..0..3..2
%K A206518 nonn,new
%O A206518 1,1
%A A206518 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206522
%S A206522 2,5,7,10,12,13,15,17,20,22,23,24,25,27,28,29,32,34,35,37,39,40,41,42,
%T A206522 45,47,49,52,55,58,59,60,62,64,66,69,71,72,76,79,80,82,83,87,89,92,94,
%U A206522 99,100,103,104,105,106,111,112,114,117,118,119,121,124,126,128,129,131,135,138,139,143,144,145,146,147,149,153,156,158,159,160,167,170,171,172,174,175
%N A206522 The pairwise differences of the terms of A004210.
%Y A206522 Cf. A004210, A206523, A206524.
%K A206522 nonn,new
%O A206522 1,1
%A A206522 N. J. A. Sloane (njas(AT)research.att.com), Feb 08 2012

%I A206517
%S A206517 43804,44350086,45142367398,45954984167446,46782397564239329,
%T A206517 47624714678427763425,48482197805424282951252,
%U A206517 49355119922405928499997192,50243759004969529025402006378,51148398036881023231239431951179
%N A206517 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206517 Column 4 of A206521
%H A206517 R. H. Hardin, <a href="/A206517/b206517.txt">Table of n, a(n) for n = 1..210</a>
%e A206517 Some solutions for n=4
%e A206517 ..0..1..0..2..0....0..0..0..0..0....0..1..1..0..0....0..1..0..0..0
%e A206517 ..3..3..0..1..3....0..1..2..1..0....0..0..1..0..0....2..3..0..3..0
%e A206517 ..2..1..1..2..1....1..3..1..3..3....0..2..3..1..1....2..0..2..2..3
%e A206517 ..1..0..2..0..3....1..1..3..0..1....2..0..1..0..0....0..0..0..3..2
%e A206517 ..0..0..2..1..1....0..0..1..0..0....0..0..3..0..1....0..2..3..2..3
%K A206517 nonn,new
%O A206517 1,1
%A A206517 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206516
%S A206516 2786,695541,177097738,45142367398,11507555684346,2933481908580803,
%T A206516 747797195466172046,190626930556386514119,48594227014894390255783,
%U A206516 12387540901864509015493628,3157806575432575464130353243
%N A206516 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206516 Column 3 of A206521
%H A206516 R. H. Hardin, <a href="/A206516/b206516.txt">Table of n, a(n) for n = 1..210</a>
%F A206516 Empirical: a(n) = 238*a(n-1) +4150*a(n-2) +44332*a(n-3) -679650*a(n-4) -12043680*a(n-5) -129978174*a(n-6) -183422200*a(n-7) -661869709*a(n-8) +1822894708*a(n-9) +3929447237*a(n-10) +24688694694*a(n-11) +16930214447*a(n-12) +31521818926*a(n-13) -66759146931*a(n-14) +41508795870*a(n-15) -100230885135*a(n-16) +89003825784*a(n-17) -52703068122*a(n-18) +46958310468*a(n-19) -22966596792*a(n-20) +7564885488*a(n-21) -2550916800*a(n-22)
%e A206516 Some solutions for n=4
%e A206516 ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..1..0....0..0..0..0
%e A206516 ..0..0..1..1....1..1..1..1....0..0..1..0....0..1..1..0....0..1..1..1
%e A206516 ..1..2..0..2....2..1..2..3....2..0..1..0....0..0..0..1....0..0..0..1
%e A206516 ..0..1..0..3....0..1..2..0....2..0..1..2....1..0..2..1....0..1..0..0
%e A206516 ..2..2..1..2....0..1..3..0....1..3..0..0....1..1..0..3....0..1..1..1
%K A206516 nonn,new
%O A206516 1,1
%A A206516 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206515
%S A206515 186,10998,695541,44350086,2830665213,180689239504,11534052523389,
%T A206515 736261766040550,46998354497911525,3000081595834713716,
%U A206515 191506483600891877185,12224578599211771061590,780340796424907303135629
%N A206515 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206515 Column 2 of A206521
%H A206515 R. H. Hardin, <a href="/A206515/b206515.txt">Table of n, a(n) for n = 1..210</a>
%F A206515 Empirical: a(n) = 66*a(n-1) -116*a(n-2) -1310*a(n-3) -7016*a(n-4) -9076*a(n-5) -9408*a(n-6) -4662*a(n-7) -4635*a(n-8) -378*a(n-9) -648*a(n-10)
%e A206515 Some solutions for n=4
%e A206515 ..0..1..0....0..1..0....0..0..1....0..1..0....0..0..0....0..1..1....0..1..0
%e A206515 ..2..0..1....1..0..0....0..1..0....0..1..0....0..0..1....0..2..1....1..0..0
%e A206515 ..0..2..0....1..0..2....2..2..1....0..1..0....2..3..2....1..1..0....0..1..1
%e A206515 ..1..2..1....3..2..1....0..3..1....0..1..1....3..1..1....3..2..3....1..0..2
%e A206515 ..3..0..0....1..0..3....0..1..0....0..0..0....1..3..3....3..0..0....3..0..1
%K A206515 nonn,new
%O A206515 1,1
%A A206515 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206514
%S A206514 15,186,2786,43804,697369,11136544,177977851,2844864002,45475406828,
%T A206514 726936444140,11620303055791,185754246374086,2969341279822087,
%U A206514 47465875753570892,758757303430379258,12128979771310068836
%N A206514 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206514 Column 1 of A206521
%H A206514 R. H. Hardin, <a href="/A206514/b206514.txt">Table of n, a(n) for n = 1..210</a>
%F A206514 Empirical: a(n) = 18*a(n-1) -27*a(n-2) -76*a(n-3) -111*a(n-4) -66*a(n-5) -24*a(n-6) for n>7
%e A206514 Some solutions for n=4
%e A206514 ..0..1....0..0....0..1....0..0....0..0....0..0....0..0....0..0....0..1....0..0
%e A206514 ..1..0....0..0....1..0....1..0....0..1....0..1....0..0....0..0....2..2....1..0
%e A206514 ..2..1....0..1....0..0....0..1....1..2....2..0....1..1....1..2....1..0....0..1
%e A206514 ..1..1....0..1....0..1....1..0....0..1....0..1....2..0....1..3....0..3....1..0
%e A206514 ..0..3....1..2....0..0....0..0....0..0....3..0....2..2....0..0....0..0....0..1
%K A206514 nonn,new
%O A206514 1,1
%A A206514 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206513
%S A206513 15,10998,177097738,45954984167446,190187445308482279868,
%T A206513 12552495871839218224290213767,13212265000747034021229857133763537610,
%U A206513 221780527293845666190504578213897949211421813469
%N A206513 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than four equal edges, and new values 0..3 introduced in row major order
%C A206513 Diagonal of A206521
%e A206513 Some solutions for n=4
%e A206513 ..0..0..0..0..1....0..0..0..0..0....0..1..0..0..0....0..0..0..0..1
%e A206513 ..2..1..3..0..0....0..1..0..2..2....2..3..0..3..0....2..3..3..0..3
%e A206513 ..1..3..2..2..0....2..2..3..2..2....2..0..2..2..3....2..2..0..1..2
%e A206513 ..2..2..1..1..0....3..1..1..2..1....0..0..0..3..2....3..2..0..1..2
%e A206513 ..1..1..2..2..1....1..2..1..3..2....0..2..3..2..3....1..1..1..0..2
%K A206513 nonn,new
%O A206513 1,1
%A A206513 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206512
%S A206512 81,549,549,3399,7737,3399,19797,93405,93405,19797,111495,1020756,
%T A206512 2105607,1020756,111495,614367,10505604,41873838,41873838,10505604,
%U A206512 614367,3337695,103812645,769739571,1483188648,769739571,103812645
%N A206512 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206512 Table starts
%C A206512 .......81........549..........3399............19797.............111495
%C A206512 ......549.......7737.........93405..........1020756...........10505604
%C A206512 .....3399......93405.......2105607.........41873838..........769739571
%C A206512 ....19797....1020756......41873838.......1483188648........47830788954
%C A206512 ...111495...10505604.....769739571......47830788954......2671430859687
%C A206512 ...614367..103812645...13397007426....1444445274765....138401623325535
%C A206512 ..3337695..997160112..224125465767...41561685861258...6781274991216915
%C A206512 .17958813.9383642739.3639506017737.1152430341911256.318257080720663863
%H A206512 R. H. Hardin, <a href="/A206512/b206512.txt">Table of n, a(n) for n = 1..112</a>
%e A206512 Some solutions for n=4 k=3
%e A206512 ..1..1..1..1....1..1..0..0....0..1..2..0....1..0..2..0....1..1..0..1
%e A206512 ..1..0..0..0....1..2..2..0....1..1..2..0....1..0..0..0....2..1..2..0
%e A206512 ..1..2..2..2....0..1..0..2....2..2..2..1....2..2..0..2....1..2..1..2
%e A206512 ..1..1..2..1....2..0..1..0....2..0..0..2....0..0..0..2....0..0..2..0
%e A206512 ..0..2..0..2....0..1..2..1....0..2..1..0....0..1..1..1....2..1..0..2
%K A206512 nonn,tabl,new
%O A206512 1,1
%A A206512 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206511
%S A206511 3337695,997160112,224125465767,41561685861258,6781274991216915,
%T A206511 1009956457230227994,140477097763717169946,18526867118888845651956,
%U A206511 2341223589530713890986871
%N A206511 Number of (n+1)X8 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206511 Column 7 of A206512
%e A206511 Some solutions for n=4
%e A206511 ..2..2..2..2..0..1..0..1....0..2..2..1..0..2..2..0....2..0..0..1..1..1..0..1
%e A206511 ..1..2..0..1..0..2..1..0....1..2..0..0..2..1..0..1....2..2..0..0..1..2..0..0
%e A206511 ..0..2..1..0..1..0..2..1....0..0..2..1..0..2..1..2....1..0..2..1..2..0..2..1
%e A206511 ..1..0..2..1..2..1..0..2....2..1..0..2..1..0..2..1....2..1..0..2..1..1..0..2
%e A206511 ..0..1..0..2..1..2..1..0....0..2..1..0..2..1..0..2....1..0..2..1..0..2..1..0
%K A206511 nonn,new
%O A206511 1,1
%A A206511 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206510
%S A206510 614367,103812645,13397007426,1444445274765,138401623325535,
%T A206510 12197461566772314,1009956457230227994,79674032690709828993,
%U A206510 6046585204361960568003,444577014495524485388130
%N A206510 Number of (n+1)X7 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206510 Column 6 of A206512
%H A206510 R. H. Hardin, <a href="/A206510/b206510.txt">Table of n, a(n) for n = 1..11</a>
%e A206510 Some solutions for n=4
%e A206510 ..0..2..2..2..1..2..0....2..1..2..2..1..2..0....1..1..0..2..1..2..2
%e A206510 ..1..2..1..0..2..0..1....2..2..2..0..0..2..1....2..2..1..0..2..1..0
%e A206510 ..2..1..0..1..0..1..2....2..0..1..2..1..0..2....0..0..2..1..0..2..1
%e A206510 ..1..2..1..2..1..2..0....0..2..2..0..2..1..0....2..1..0..2..1..0..2
%e A206510 ..1..0..2..0..2..1..2....2..1..0..1..0..2..1....1..0..1..0..2..1..0
%K A206510 nonn,new
%O A206510 1,1
%A A206510 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206509
%S A206509 111495,10505604,769739571,47830788954,2671430859687,138401623325535,
%T A206509 6781274991216915,318257080720663863,14433205926606629808,
%U A206509 636578185379733865623,27437199789684970128768,1159984309083884155349523
%N A206509 Number of (n+1)X6 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206509 Column 5 of A206512
%H A206509 R. H. Hardin, <a href="/A206509/b206509.txt">Table of n, a(n) for n = 1..122</a>
%e A206509 Some solutions for n=4
%e A206509 ..1..2..1..2..1..0....2..1..0..1..1..1....0..0..2..0..0..2....1..0..2..1..2..1
%e A206509 ..2..1..2..1..2..1....0..1..0..1..2..1....2..2..1..2..1..0....2..2..1..2..0..2
%e A206509 ..1..2..1..2..1..0....0..2..1..2..1..2....0..1..0..1..2..1....2..0..2..0..2..1
%e A206509 ..1..0..2..1..2..1....1..0..2..0..2..0....2..2..1..0..1..0....2..1..0..1..0..2
%e A206509 ..2..1..0..2..1..0....2..1..0..1..0..1....1..0..2..1..0..2....0..2..1..0..1..0
%K A206509 nonn,new
%O A206509 1,1
%A A206509 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206508
%S A206508 19797,1020756,41873838,1483188648,47830788954,1444445274765,
%T A206508 41561685861258,1152430341911256,31037137973162439,816543046767301710,
%U A206508 21075169065968338644,535422905459574902352,13424260728041018703732
%N A206508 Number of (n+1)X5 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206508 Column 4 of A206512
%H A206508 R. H. Hardin, <a href="/A206508/b206508.txt">Table of n, a(n) for n = 1..210</a>
%e A206508 Some solutions for n=4
%e A206508 ..2..1..2..1..0....2..2..0..0..1....2..1..2..2..2....1..0..2..0..1
%e A206508 ..1..2..1..2..1....1..2..0..1..2....2..1..1..1..1....2..0..1..0..2
%e A206508 ..1..0..2..0..2....1..2..2..0..0....0..0..0..0..1....2..1..2..1..0
%e A206508 ..2..1..0..1..0....0..1..0..2..1....1..1..0..2..1....0..2..0..2..1
%e A206508 ..0..2..1..2..1....1..2..1..0..2....2..1..1..1..0....1..0..1..0..2
%K A206508 nonn,new
%O A206508 1,1
%A A206508 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206507
%S A206507 3399,93405,2105607,41873838,769739571,13397007426,224125465767,
%T A206507 3639506017737,57759037910145,900227925756975,13830131604178527,
%U A206507 210015057307677960,3159110458844182341,47153204170712651499
%N A206507 Number of (n+1)X4 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206507 Column 3 of A206512
%H A206507 R. H. Hardin, <a href="/A206507/b206507.txt">Table of n, a(n) for n = 1..210</a>
%F A206507 Empirical: a(n) = 43*a(n-1) -560*a(n-2) -301*a(n-3) +58140*a(n-4) -266678*a(n-5) -2052111*a(n-6) +15616295*a(n-7) +32553554*a(n-8) -428019493*a(n-9) -140101975*a(n-10) +7157244486*a(n-11) -3878496130*a(n-12) -81350820700*a(n-13) +92282193531*a(n-14) +661037913951*a(n-15) -1153139822216*a(n-16) -3750375673397*a(n-17) +10272007050117*a(n-18) +11307559698334*a(n-19) -66861682621412*a(n-20) +30396309007373*a(n-21) +286797532029361*a(n-22) -577253429355795*a(n-23) -515274276475600*a(n-24) +3288213731014612*a(n-25) -2319778381651637*a(n-26) -9210618806887255*a(n-27) +21522479749004936*a(n-28) +386994035523308*a(n-29) -84485292746124261*a(n-30) +120213656794023933*a(n-31) +200479767563622238*a(n-32) -651342756788935634*a(n-33) -218995076753278783*a(n-34) +2240168025280749856*a(n-35) -562599124935184162*a(n-36) -5718074917398148106*a(n-37) +3896739065818876813*a(n-38) +11157819107569565092*a(n-39) -12344609689373548452*a(n-40) -16519275709140140613*a(n-41) +27575879497435303388*a(n-42) +17037875789330156165*a(n-43) -47145213836213708945*a(n-44) -7362526243707437385*a(n-45) +61857433515558819146*a(n-46) -12108244633773880734*a(n-47) -59603119051405911044*a(n-48) +29684960133920835544*a(n-49) +38742581976811825635*a(n-50) -30923987787997544536*a(n-51) -14971734351034264399*a(n-52) +17593615001634785102*a(n-53) +3693415397812903566*a(n-54) -5865717176480173312*a(n-55) -2285287018681816803*a(n-56) +3279660163779884229*a(n-57) +1117819441930804736*a(n-58) -3810709135189499181*a(n-59) +1209355289383345147*a(n-60) +2609574092234205911*a(n-61) -2025943866398526978*a(n-62) -748404131952133991*a(n-63) +1374111095557955776*a(n-64) -254509843997104061*a(n-65) -529334312705599289*a(n-66) +396629471902305293*a(n-67) +58155671568981384*a(n-68) -197049151694905447*a(n-69) +56626096310808263*a(n-70) +40736308817253334*a(n-71) -33420664239714004*a(n-72) +3623368762119517*a(n-73) +9036297022738612*a(n-74) -4794775820304365*a(n-75) -1345199351985969*a(n-76) +1724273351995858*a(n-77) -167082321416272*a(n-78) -345131182064849*a(n-79) +193604547939728*a(n-80) -1504243529745*a(n-81) -48820505858969*a(n-82) +17081169883351*a(n-83) +2777154345220*a(n-84) -2638202409776*a(n-85) +598690007258*a(n-86) -61405109728*a(n-87) -59373700630*a(n-88) +37856079048*a(n-89) -5347077919*a(n-90) -1361201971*a(n-91) +736710444*a(n-92) -144960710*a(n-93) -4344552*a(n-94) +7376088*a(n-95) -1422720*a(n-96) +99840*a(n-97)
%e A206507 Some solutions for n=4
%e A206507 ..1..0..1..2....1..1..1..1....1..1..1..2....1..2..0..0....0..1..2..1
%e A206507 ..2..1..0..1....1..0..0..0....1..2..2..2....2..0..2..1....0..0..0..2
%e A206507 ..1..0..1..2....1..2..2..2....1..1..2..0....2..1..0..2....2..2..1..0
%e A206507 ..0..1..0..0....1..1..2..1....1..2..0..1....0..2..1..0....1..1..2..1
%e A206507 ..1..0..2..1....0..2..0..2....1..2..1..2....1..0..2..1....1..2..0..2
%K A206507 nonn,new
%O A206507 1,1
%A A206507 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206506
%S A206506 549,7737,93405,1020756,10505604,103812645,997160112,9383642739,
%T A206506 86977382523,797102746356,7242496383288,65374662519669,
%U A206506 587135801448765,5252624153233296,46849534683131463,416887973306651277
%N A206506 Number of (n+1)X3 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206506 Column 2 of A206512
%H A206506 R. H. Hardin, <a href="/A206506/b206506.txt">Table of n, a(n) for n = 1..210</a>
%F A206506 Empirical: a(n) = 20*a(n-1) -108*a(n-2) -116*a(n-3) +2081*a(n-4) -2052*a(n-5) -9023*a(n-6) +14684*a(n-7) +4103*a(n-8) -20511*a(n-9) +25590*a(n-10) -24138*a(n-11) -683*a(n-12) +30616*a(n-13) -32567*a(n-14) +8765*a(n-15) -850*a(n-16) +4935*a(n-17) +9392*a(n-18) -14485*a(n-19) +4202*a(n-20) +738*a(n-21) -1528*a(n-22) +1448*a(n-23) -608*a(n-24) +96*a(n-25)
%e A206506 Some solutions for n=4
%e A206506 ..0..0..2....2..2..0....0..0..2....2..0..1....0..0..2....2..1..0....2..1..2
%e A206506 ..1..2..0....1..2..0....0..0..0....2..1..2....0..0..2....2..1..2....1..2..1
%e A206506 ..1..2..1....1..1..2....1..2..0....0..2..0....2..2..1....2..1..1....0..0..2
%e A206506 ..2..0..2....1..0..1....1..2..2....2..0..1....1..0..2....0..1..2....2..1..0
%e A206506 ..2..1..0....1..2..2....0..1..0....0..1..0....2..1..0....0..0..2....0..2..1
%K A206506 nonn,new
%O A206506 1,1
%A A206506 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206505
%S A206505 81,549,3399,19797,111495,614367,3337695,17958813,95989473,510679479,
%T A206505 2707979133,14326037895,75662455155,399129270627,2103642300213,
%U A206505 11080520280189,58338276097647,307048060430277,1615687942238055
%N A206505 Number of (n+1)X2 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206505 Column 1 of A206512
%H A206505 R. H. Hardin, <a href="/A206505/b206505.txt">Table of n, a(n) for n = 1..210</a>
%F A206505 Empirical: a(n) = 9*a(n-1) -18*a(n-2) -15*a(n-3) +36*a(n-4) -20*a(n-5) +11*a(n-6) -2*a(n-7)
%e A206505 Some solutions for n=4
%e A206505 ..2..2....1..2....1..2....1..2....2..2....2..1....2..1....0..0....1..2....2..0
%e A206505 ..1..0....1..2....0..1....1..2....2..1....2..0....0..2....2..1....2..2....2..2
%e A206505 ..0..1....2..0....2..2....1..0....2..1....0..1....1..0....0..2....1..2....0..2
%e A206505 ..1..2....0..1....1..0....2..1....1..1....0..2....2..1....2..0....1..0....2..0
%e A206505 ..0..2....2..2....2..1....1..2....2..1....1..0....1..2....2..1....2..1....1..2
%K A206505 nonn,new
%O A206505 1,1
%A A206505 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206504
%S A206504 81,7737,2105607,1483188648,2671430859687,12197461566772314,
%T A206504 140477097763717169946
%N A206504 Number of (n+1)X(n+1) 0..2 arrays with the number of clockwise edge increases in 2X2 subblocks nondecreasing rightwards and downwards
%C A206504 Diagonal of A206512
%e A206504 Some solutions for n=4
%e A206504 ..1..2..0..1..1....1..0..2..1..1....0..2..1..2..1....2..0..1..0..1
%e A206504 ..2..1..1..0..2....1..2..1..0..2....1..1..1..2..0....2..0..2..1..0
%e A206504 ..1..0..2..1..0....1..1..0..1..0....1..2..2..1..1....2..1..0..2..1
%e A206504 ..2..1..0..2..1....2..2..1..2..1....1..1..1..0..2....0..2..1..0..2
%e A206504 ..0..2..1..0..2....0..1..2..1..0....0..0..2..1..0....1..0..2..1..0
%K A206504 nonn,new
%O A206504 1,1
%A A206504 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206503
%S A206503 1,1,1,1,1,1,2,4,4,2,5,34,65,34,5,15,481,3130,3130,481,15,52,8731,
%T A206503 178998,502599,178998,8731,52,203,174454,10502458,83045488,83045488,
%U A206503 10502458,174454,203,876,3603244,618242826,13752277898,38638317841,13752277898
%N A206503 T(n,k)=Number of nXk 0..7 arrays with no element equal to another within a city block distance of two, and new values 0..7 introduced in row major order
%C A206503 Table starts
%C A206503 ...1.......1.........1...........2...........5..........15............52
%C A206503 ...1.......1.........4..........34.........481........8731........174454
%C A206503 ...1.......4........65........3130......178998....10502458.....618242826
%C A206503 ...2......34......3130......502599....83045488.13752277898.2277207309720
%C A206503 ...5.....481....178998....83045488.38638317841
%C A206503 ..15....8731..10502458.13752277898
%C A206503 ..52..174454.618242826
%C A206503 .203.3603244
%H A206503 R. H. Hardin, <a href="/A206503/b206503.txt">Table of n, a(n) for n = 1..49</a>
%e A206503 Some solutions for n=4 k=3
%e A206503 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206503 ..2..3..0....3..4..5....2..3..0....2..3..4....2..3..0....2..3..0....3..4..0
%e A206503 ..1..4..5....5..2..1....1..4..5....4..5..6....4..5..6....4..5..1....2..5..1
%e A206503 ..5..0..1....6..7..3....5..2..1....6..1..7....6..7..1....3..0..6....0..3..6
%Y A206503 Column 1 is A056273(n-2)
%Y A206503 Column 2 is A198976(n-1)
%K A206503 nonn,tabl,new
%O A206503 1,7
%A A206503 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206502
%S A206502 5,481,178998,83045488,38638317841
%N A206502 Number of nX5 0..7 arrays with no element equal to another within a city block distance of two, and new values 0..7 introduced in row major order
%C A206502 Column 5 of A206503
%e A206502 Some solutions for n=4
%e A206502 ..0..1..2..3..0....0..1..2..0..1....0..1..2..0..1....0..1..2..3..0
%e A206502 ..2..3..0..1..2....2..3..4..5..6....2..3..4..5..2....2..3..0..1..2
%e A206502 ..4..5..6..4..7....1..5..7..1..2....4..0..1..6..3....4..5..6..7..3
%e A206502 ..0..7..1..5..3....0..2..3..0..4....1..2..3..7..4....1..7..2..0..4
%K A206502 nonn,new
%O A206502 1,1
%A A206502 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206501
%S A206501 2,34,3130,502599,83045488,13752277898,2277207309720,377099217207514,
%T A206501 62445595052884887,10340702774554689140,1712370276674504142791,
%U A206501 283560339737534813492944
%N A206501 Number of nX4 0..7 arrays with no element equal to another within a city block distance of two, and new values 0..7 introduced in row major order
%C A206501 Column 4 of A206503
%e A206501 Some solutions for n=4
%e A206501 ..0..1..2..0....0..1..2..0....0..1..2..0....0..1..2..0....0..1..2..0
%e A206501 ..2..3..4..5....2..3..4..5....2..3..4..5....2..3..4..1....2..3..4..1
%e A206501 ..4..5..6..3....1..0..6..1....4..0..6..1....4..5..6..7....1..5..6..7
%e A206501 ..6..7..1..4....5..4..2..3....1..2..3..7....7..0..3..2....3..4..1..0
%K A206501 nonn,new
%O A206501 1,1
%A A206501 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206500
%S A206500 1,4,65,3130,178998,10502458,618242826,36410671102,2144479610070,
%T A206500 126304554589462,7439032094765766,438141046109721622,
%U A206500 25805450786659366998,1519878816318690262678,89517196729004371783830
%N A206500 Number of nX3 0..7 arrays with no element equal to another within a city block distance of two, and new values 0..7 introduced in row major order
%C A206500 Column 3 of A206503
%H A206500 R. H. Hardin, <a href="/A206500/b206500.txt">Table of n, a(n) for n = 1..62</a>
%F A206500 Empirical: a(n) = 55*a(n-1) +370*a(n-2) -8556*a(n-3) +15608*a(n-4) +70028*a(n-5) -175808*a(n-6) +98304*a(n-7) for n>9
%e A206500 Some solutions for n=4
%e A206500 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206500 ..2..3..0....2..3..0....2..3..0....3..4..0....2..3..4....2..3..4....2..3..4
%e A206500 ..1..4..5....4..5..1....1..4..5....2..5..1....1..0..5....1..5..6....4..5..1
%e A206500 ..5..0..1....1..0..4....5..2..1....0..3..6....3..4..2....3..4..7....6..2..3
%K A206500 nonn,new
%O A206500 1,2
%A A206500 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A185303
%S A185303 0,0,0,0,0,1,100,967255,14971125930
%N A185303 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 5.
%H A185303 L. Travis, <a href="http://arXiv.org/abs/math.CO/9811127">[math/9811127] Graphical Enumeration: A Species-Theoretic Approach</a>
%Y A185303 Cf. A001373, A129524, A185193, A185194.
%K A185303 nonn,more,new
%O A185303 1,7
%A A185303 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 08 2012

%I A185194
%S A185194 0,0,0,0,1,40,35317,56150820,111359017198
%N A185194 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 4.
%H A185194 L. Travis, <a href="http://arXiv.org/abs/math.CO/9811127">[math/9811127] Graphical Enumeration: A Species-Theoretic Approach</a>
%Y A185194 Cf. A001373, A129524, A185193, A185303.
%K A185194 nonn,more,new
%O A185194 1,6
%A A185194 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 08 2012

%I A185193
%S A185193 0,0,0,1,13,1499,257290,56150820,14971125930
%N A185193 Number of unlabeled digraphs on n vertices such that every vertex has outdegree 3.
%H A185193 L. Travis, <a href="http://arXiv.org/abs/math.CO/9811127">[math/9811127] Graphical Enumeration: A Species-Theoretic Approach</a>
%Y A185193 Cf. A001373, A129524, A185194, A185303.
%K A185193 nonn,more,new
%O A185193 1,5
%A A185193 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 08 2012

%I A205643
%S A205643 2,84,480,32256,58254,61440,1556480,3932160,1505806848,107870873600
%N A205643 (2,k)-perfect numbers (A019278) such that the next (2,k)-perfect number has the same value of k (in A098223)
%Y A205643 Cf. A019278, A098223
%K A205643 nonn,new
%O A205643 1,1
%A A205643 Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Feb 08 2012

%I A205597
%S A205597 1,15,21,1023,29127,550095,355744082763
%N A205597 Odd terms of A019278: odd n such that sigma(sigma(n))/n is an integer.
%e A205597 15 is odd, sigma(15) = 24, sigma(24) = 60, and 60/15 is integer.
%Y A205597 Cf. A019278, A000302.
%K A205597 nonn,new
%O A205597 1,2
%A A205597 Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), Feb 08 2012

%I A185331
%S A185331 1,1,1,0,1,1,1,1,1,1,0,2,2,1,1,1,1,3,3,1,1,0,3,3,4,4,1,1,
%T A185331 1,1,6,6,5,5,1,1,0,4,4,10,10,6,6,1,1,1,1,10,10,15,15,7,7,
%U A185331 1,1,0,5,5,20,20,21,21,8,8,1,1
%V A185331 1,-1,1,0,-1,1,1,-1,-1,1,0,2,-2,-1,1,-1,1,3,-3,-1,1,0,-3,3,4,-4,-1,1,
%W A185331 1,-1,-6,6,5,-5,-1,1,0,4,-4,-10,10,6,-6,-1,1,-1,1,10,-10,-15,15,7,-7,
%X A185331 -1,1,0,-5,5,20,-20,-21,21,8,-8,-1,1
%N A185331 Riordan array ((1-x+x^2)/(1+x^2), x/(1+x^2)).
%C A185331 Triangle T(n,k), read by rows, given by (-1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%F A185331 T(n,k) = T(n-1,k-1) - T(n-2,k), T(0,0) = 1, T(0,1) = -1, T(0,2) = 0.
%F A185331 G.f.: (1-x+x^2)/(1-y*x+x^2).
%F A185331 Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A184334(n), A163805(n), A000007(n), A028310(n), A025169(n-1), A005320(n) (n>0) for x = -1, 0, 1, 2, 3, 4 respectively.
%F A185331 T(n,n) = 1, T(n+1,n) = -1, T(n+2,n) = -n, T(n+3,n) = n+1, T(n+4,n) = n(n+1)/2 = A000217(n).
%F A185331 T(2n,2k) = (-1)^(n-k) * A128908(n,k), T(2n+1,2k+1) = -T(2n+1,2k) = A129818(n,k), T(2n+2,2k+1) = (-1)*A053122(n,k). - DELEHAM Philippe, Feb 09 2012
%e A185331 Triangle begins :
%e A185331 1
%e A185331 -1, 1
%e A185331 0, -1, 1
%e A185331 1, -1, -1, 1
%e A185331 0, 2, -2, -1, 1
%e A185331 -1, 1, 3, -3, -1, 1
%e A185331 0, -3, 3, 4, -4, -1, 1
%e A185331 1, -1, -6, 6, 5, -5, -1, 1
%e A185331 0, 4, -4, -10, 10, 6, -6, -1, 1
%e A185331 -1, 1, 10, -10, -15, 15, 7, -7, -1, 1
%e A185331 0, -5, 5, 20, -20, -21, 21, 8, -8, -1, 1
%e A185331 1, -1, -15, 15, 35, -35, -28, 28, 9, -9, -1, 1
%Y A185331 Cf. A206474 (unsigned version)
%Y A185331 Cf. A007318, A053122, A078812, A085478, A128908, A129818,
%K A185331 easy,sign,tabl,new
%O A185331 0,12
%A A185331 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 08 2012

%I A206399
%S A206399 1,43,166,371,658,1027,1478,2011,2626,3323,4102,4963,5906,6931,8038,
%T A206399 9227,10498,11851,13286,14803,16402,18083,19846,21691,23618,25627,
%U A206399 27718,29891,32146,34483,36902,39403,41986,44651,47398,50227,53138,56131,59206,62363
%N A206399 a(0) = 1, a(n) = 41*n^2 + 2 for n>0.
%C A206399 Apart from the first term, numbers of the form (r^2+2*s^2)*n^2+2 = (r*n)^2+(s*n-1)^2+(s*n+1)^2: in this case is r=3, s=4. After 1, all terms are in A000408.
%H A206399 Bruno Berselli, <a href="/A206399/b206399.txt">Table of n, a(n) for n = 0..1000</a>
%H A206399 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A206399 G.f.: (1+x)*(1+39*x+x^2)/(1-x)^3.
%t A206399 Join[{1}, 41 Range[39]^2 + 2]
%o A206399 (MAGMA) [n eq 0 select 1 else 41*n^2+2: n in [0..39]];
%Y A206399 Sequences of the same type: A005893, A005897, A005899, A005901, A005903, A005905, A005914, A005918, A005919, A008527, A010000-A010023.
%K A206399 nonn,easy,new
%O A206399 0,2
%A A206399 Bruno Berselli (berselli.bruno(AT)yahoo.it), Feb 07 2012

%I A206475
%S A206475 0,1,1,1,2,4,1,2,3,6,4,6,6,2,3,5,9,11,6,0,2,12,12,11,9,8,2,
%T A206475 10,20,22,9,1,4,8,3,15,18,6,4,20,28,30,12,2,6,24,24,21,
%U A206475 22,11,4,16,32,20,10,6,8,30,34,36,30,12,1,5,28,46
%V A206475 0,1,1,1,-2,4,-1,2,-3,6,-4,6,-6,2,3,5,-9,11,-6,0,-2,12,-12,11,-9,8,-2,
%W A206475 10,-20,22,-9,-1,-4,8,-3,15,-18,6,-4,20,-28,30,-12,-2,-6,24,-24,21,
%X A206475 -22,11,4,16,-32,20,-10,6,-8,30,-34,36,-30,12,1,5,-28,46
%N A206475 First differences of A206369.
%C A206475 a(A206368(n)) = 0.
%H A206475 Reinhard Zumkeller, <a href="/A206475/b206475.txt">Table of n, a(n) for n = 1..10000</a>
%F A206475 a(n) = A206369(n+1) - A206369(n).
%o A206475 (Haskell)
%o A206475 a206475 n = a206475_list !! (n-1)
%o A206475 a206475_list = zipWith (-) (tail a206369_list) a206369_list
%K A206475 sign,new
%O A206475 1,5
%A A206475 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 08 2012

%I A206481
%S A206481 0,1,1,7,26,57,99,159,244,353,485,647,846,1081,1351,1663,2024,2433,
%T A206481 2889,3399,3970,4601,5291,6047,6876,7777,8749,9799,10934,12153,13455,
%U A206481 14847,16336,17921,19601,21383,23274,25273,27379,29599,31940,34401,36981,39687
%N A206481 a(n) + a(n+2) = n^3.
%H A206481 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).
%t A206481 LinearRecurrence[{4, -7, 8, -7, 4, -1}, {0, 1, 1, 7, 26, 57}, 60]
%Y A206481 Cf. A144129 (bisection).
%K A206481 nonn,easy,new
%O A206481 1,4
%A A206481 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 08 2012

%I A204325
%S A204325 16,8,7,7,6,7,5,6,8,6,9,9,7,6,8,10,8,9,9,6,8,7,8,11,11,8,7,4,3,12,11,
%T A204325 12,9,14,10,11,12,11,11,12,9,13,10,8,5,11,18,16,12,11,11,8,12,12,12,
%U A204325 13,9,9,7,4,8,16,14,10,8,16,16,20,16
%N A204325 Residual of an asymptotic formula for the n-th prime: Floor(prime(n)-n*log(n) + n - n*log(log(n)) - (n/log(n))*(log(log(n)) - 2) + (log(log(n)) - 6)*n*log(log(n))/(2*log(n)^2)).
%C A204325 prime(n) ~ n*log(n)+n-n*log(log(n)) - (n/log(n))*(log(log(n))-2)+(log(log(n))-6)*n*log(log(n))/(2*log(n)^2))
%D A204325 M. Cipolla, "La determinazione asintotica dell'n-mo numero primo.", Rend. d. R. Acc. di sc. fis. e mat. di Napoli, s. 3, VIII (1902), pp. 132-166.
%H A204325 Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442">Estimates of Some Functions Over Primes without R.H.</a> (2010).
%H A204325 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>
%t A204325 Table[Floor[Prime[n]-n*Log[n]+n-n*Log[Log[n]]- (n/Log[n]) (Log[Log[n]]-2)+(Log[Log[n]]-6)*n*Log[Log[n]]/(2*Log[n]^2)],{n,2,100}]
%Y A204325 Cf. A064658, A059111.
%K A204325 nonn,new
%O A204325 2,1
%A A204325 José María Grau Ribas (grau.ribas(AT)gmail.com), Jan 14 2012

%I A206477
%S A206477 3,3,36,76,690,2996,22368,147472,1284653,11006509
%N A206477 (Number of permutations of {1,2,...,n} for which sums of three consecutive numbers (with wraparound) are all distinct)/2n
%C A206477 Essentially different ways 1,2,...,n can be placed around a circle so that sums of three consecutive numbers are distinct.
%e A206477 a(4)=3 because up to rotation/reflection the only three permutations which work are 1234, 1243, and 1324
%Y A206477 Cf. A040018
%K A206477 hard,nonn,new
%O A206477 4,1
%A A206477 Steve Butler (butler(AT)iastate.edu), Feb 08 2012

%I A206480
%S A206480 60,46,690,2529,67636,184014,2374017,19353643
%N A206480 (Number of permutations of {1,2,...,n} for which sums of five consecutive numbers (with wraparound) are all distinct)/2n
%C A206480 Essentially different ways 1,2,3,...n can be placed around a circle so that sums of five consecutive numbers are distinct.
%e A206480 Some examples of such permutations are: 12568374; 13756824; and 16253847
%Y A206480 Cf. A040018
%K A206480 nonn,new
%O A206480 6,1
%A A206480 Steve Butler (butler(AT)iastate.edu), Feb 08 2012

%I A206478
%S A206478 12,8,76,694,2529,23679,177885,1482021,14666021
%N A206478 (Number of permutations of {1,2,...,n} for which sums of four consecutive numbers (with wraparound) are all distinct)/2n
%C A206478 Essentially different ways that 1,2,...,n can be placed around a circle so that the sums of three consecutive terms are all distinct.
%e A206478 a(6)=8 because up to rotation/reflection the only three permutations which work are 123465, 123546, 124653, 125364, 126543, 132456, 132645, 136524.
%Y A206478 CF A040018
%K A206478 hard,nonn,new
%O A206478 5,1
%A A206478 Steve Butler (butler(AT)iastate.edu), Feb 08 2012

%I A206474
%S A206474 1,1,1,0,1,1,1,1,1,1,0,2,2,1,1,1,1,3,3,1,1,0,3,3,4,4,1,1,1,1,6,6,5,5,
%T A206474 1,1,0,4,4,10,10,6,6,1,1,1,1,10,10,15,15,7,7,1,1,0,5,5,20,20,21,21,8,
%U A206474 8,1,1,1,1,15,15,35,35,28,28,9,9,1,1
%N A206474 Riordan array ((1+x-x^2)/(1-x^2), x/(1-x^2)).
%C A206474 Triangle T(n,k), read by rows, given by (1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A206474 Antidiagonal sums are A158780(n+1).
%C A206474 Row sums are 2*Fibonacci(n) = 2*A000045(n), n>0.
%F A206474 T(2n, 2k) = A128908(n,k), T(2n+1, 2k) = T(2n+1, 2k+1) = A085478(n,k) = Binomial (n+k, 2k), T(2n+2, 2k+1) = A078812(n,k) = Binomial(n+k-1, 2k-1).
%F A206474 T(n,k) = T(n-1,k-1) + T(n-2,k), T(0,0) = T(0,1) = 1, T(0,2) = 0.
%F A206474 G.f.: (1+x-x^2)/(1-x*y-x^2).
%F A206474 Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^(n-1)* A141682(n-1), A000007(n), A135528(n-1), A055389(n) for x = -2, -1, 0, 1 respectively .
%e A206474 Triangle begins :
%e A206474 1
%e A206474 1, 1
%e A206474 0, 1, 1
%e A206474 1, 1, 1, 1
%e A206474 0, 2, 2, 1, 1
%e A206474 1, 1, 3, 3, 1, 1
%e A206474 0, 3, 3, 4, 4, 1, 1
%e A206474 1, 1, 6, 6, 5, 5, 1, 1
%e A206474 0, 4, 4, 10, 10, 6, 6, 1, 1
%e A206474 1, 1, 10, 10, 15, 15, 7, 7, 1, 1
%e A206474 0, 5, 5, 20, 20, 21, 21, 8, 8, 1, 1
%e A206474 1, 1, 15, 15, 35, 35, 28, 28, 9, 9, 1, 1
%Y A206474 Cf. A007318, A078812, A085478, A128908,
%K A206474 easy,nonn,tabl,new
%O A206474 0,12
%A A206474 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 08 2012

%I A206424
%S A206424 1,2,2,2,4,4,2,4,5,2,4,4,4,8,8,4,8,10,2,4,5,4,8,10,5,10,14,2,4,4,4,8,
%T A206424 8,4,8,10,4,8,8,8,16,16,8,16,20,4,8,10,8,16,20,10,20,28,2,4,5,4,8,10,
%U A206424 5,10,14,4,8,10,8,16,20,10,20,28,5,10,14,10,20,28
%N A206424 The number of 1's in row n of Pascal's Triangle (mod 3)
%C A206424 A006047(n) = a(n) + A206425(n)
%H A206424 Marcus Jaiclin, <a href="http://www.westfield.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle, Mod 2,3,5</a>
%e A206424 Example: Rows 0-8 of Pascal's Triangle (mod 3) are:
%e A206424 1                   So a(0) = 1
%e A206424 1 1                 So a(1) = 2
%e A206424 1 2 1               So a(2) = 2
%e A206424 1 0 0 1                 .
%e A206424 1 1 0 1 1               .
%e A206424 1 2 1 1 2 1             .
%e A206424 1 0 0 2 0 0 1
%e A206424 1 1 0 2 2 0 1 1
%e A206424 1 2 1 2 1 2 1 2 1
%t A206424 Table[Count[Mod[Binomial[n, Range[0, n]], 3], 1], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)
%Y A206424 Cf. A083093, A062296, A006047.
%K A206424 nonn,easy,new
%O A206424 0,2
%A A206424 Marcus Jaiclin (mjaiclin(AT)westfield.ma.edu), Feb 07 2012

%I A206425
%S A206425 0,0,1,0,0,2,1,2,4,0,0,2,0,0,4,2,4,8,1,2,4,2,4,8,1,2,4,2,4,8,4,8,13,0,
%T A206425 0,2,0,0,4,2,4,8,0,0,4,0,0,8,4,8,16,2,4,8,4,8,16,8,16,26,1,2,4,2,4,8,
%U A206425 4,8,13,2,4,8,4,8,16,8,16,26,4,8,13,8,16,26
%N A206425 The number of 2's in row n of Pascal's Triangle (mod 3)
%C A206425 A006047(n) = A206424(n) + a(n)
%H A206425 Marcus Jaiclin, <a href="http://www.westfield.ma.edu/math/faculty/jaiclin/writings/research/pascals_triangle/">Pascal's Triangle, Mod 2,3,5</a>
%e A206425 Example: Rows 0-8 of Pascal's Triangle (mod 3) are:
%e A206425 1                   So a(0) = 0
%e A206425 1 1                 So a(1) = 0
%e A206425 1 2 1               So a(2) = 1
%e A206425 1 0 0 1                 .
%e A206425 1 1 0 1 1               .
%e A206425 1 2 1 1 2 1             .
%e A206425 1 0 0 2 0 0 1
%e A206425 1 1 0 2 2 0 1 1
%e A206425 1 2 1 2 1 2 1 2 1
%t A206425 Table[Count[Mod[Binomial[n, Range[0, n]], 3], 2], {n, 0, 99}] (* Alonso del Arte, Feb 07 2012 *)
%Y A206425 Cf. A083093, A062296, A006047.
%K A206425 nonn,easy,new
%O A206425 0,6
%A A206425 Marcus Jaiclin (mjaiclin(AT)westfield.ma.edu), Feb 07 2012

%I A206429
%S A206429 2,6,3,36,24,4,320,240,60,5,3750,3000,900,120,6,54432,45360,15120,
%T A206429 2520,210,7,941192,806736,288120,54880,5880,336,8,18874368,16515072,
%U A206429 6193152,1290240,161280,12096,504,9,430467210,382637520,148803480,33067440,4592700,408240,22680,720
%N A206429 Triangular array read by rows. T(n,k) is the number of rooted lablelled trees on n nodes such that the root node has degree k. n>=2, 1<=k<=n-1
%C A206429 Row sums = A000169
%C A206429 Column 1 = A055541
%e A206429 2
%e A206429   6     3
%e A206429   36    24    4
%e A206429   320   240   60    5
%e A206429   3750  3000  900   120   6
%e A206429   54432 45360 15120 2520  210  7
%t A206429 nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Drop[Transpose[Table[Range[0,nn]!CoefficientList[Series[x t^k/k!,{x,0,nn}],x],{k,1,8}]],2]]//Flatten
%K A206429 nonn,new
%O A206429 2,1
%A A206429 Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Feb 07 2012

%I A206329
%S A206329 5,30,42,78,138,186,210,222,258,330,390,410,434,462,618,762,786,798,
%T A206329 906,930,946,966,978,1002,1030,1230,1290,1334,1374,1410,1446,1482,
%U A206329 1518,1542,1606,1722,1758,1770,1794,1830,1866,1878,1938,1974,2006,2022,2190,2226
%N A206329 Squarefree sums of 2 successive primes.
%C A206329 Intersection of A001043 and A005117, both infinite, but is their intersection infinite?
%C A206329 Also note that the only prime is a(1)=5 and there are no semiprimes (products of 2 primes A001358).
%H A206329 Charles R Greathouse IV, <a href="/A206329/b206329.txt">Table of n, a(n) for n = 1..10000</a>
%e A206329 a(1)=5=A001043(1)=A005117(4), a(2)=30=A001043(6)=A005117(19), a(3)=42=A001043(8)=A005117(28).
%o A206329 (PARI) p=2;forprime(q=3,1e4,if(issquarefree(p+q),print1(p+q", "));p=q) \\ Charles R Greathouse IV, Feb 08 2012
%Y A206329 Cf. A001043, A001358, A005117, A206462.
%K A206329 nonn,new
%O A206329 1,1
%A A206329 Moshe Levin (moshe.levin(AT)mail.ru), Feb 06 2012

%I A206462
%S A206462 2,13,19,37,67,89,103,109,127,163,193,199,211,229,307,379,389,397,449,
%T A206462 463,467,479,487,499,509,613,643,661,683,701,719,739,757,769,797,859,
%U A206462 877,883,887,911,929,937,967,983,997,1009,1093,1109,1163,1201,1237,1279
%N A206462 Primes p such that p + nextprime(p) is a squarefree number (A005117).
%H A206462 Reinhard Zumkeller, <a href="/A206462/b206462.txt">Table of n, a(n) for n = 1..10000</a>
%F A206462 a(n) + nextprime(a(n)) = A206329(n).
%F A206462 A008966(A001043(A049084(a(n)))) = 1. [Reinhard Zumkeller, Feb 08 2012]
%e A206462 13 + 17 = 30 = A206329(2).
%t A206462 Prime[Select[Range[200], Abs[MoebiusMu[Prime[#] + Prime[# + 1]]] == 1 &]] (* Alonso del Arte, Feb 08 2012 *)
%o A206462 (Haskell)
%o A206462 a206462 n = a206462_list !! (n-1)
%o A206462 a206462_list = map (a000040 . (+ 1)) $
%o A206462                    elemIndices 1 $ map a008966 a001043_list
%o A206462 -- Reinhard Zumkeller, Feb 08 2012
%Y A206462 Cf. A001043, A001358, A005117, A206329.
%K A206462 nonn,new
%O A206462 1,1
%A A206462 Moshe Levin (moshe.levin(AT)mail.ru), Feb 07 2012

%I A206472
%S A206472 25,121,121,441,1411,441,1849,11025,11025,1849,7225,106891,110889,
%T A206472 106891,7225,29241,958441,1896129,1896129,958441,29241,116281,8963667,
%U A206472 23707161,68352739,23707161,8963667,116281,466489,82609921,356190129,1693734025
%N A206472 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having zero permanent
%C A206472 Table starts
%C A206472 .....25.......121.........441...........1849.............7225
%C A206472 ....121......1411.......11025.........106891...........958441
%C A206472 ....441.....11025......110889........1896129.........23707161
%C A206472 ...1849....106891.....1896129.......68352739.......1693734025
%C A206472 ...7225....958441....23707161.....1693734025......61244865529
%C A206472 ..29241...8963667...356190129....52552339515....3420616959081
%C A206472 .116281..82609921..4803737481..1457044540561..144660587865049
%C A206472 .466489.767387611.69030731169.43214597865811.7310031634630729
%H A206472 R. H. Hardin, <a href="/A206472/b206472.txt">Table of n, a(n) for n = 1..684</a>
%e A206472 Some solutions for n=4 k=3
%e A206472 ..0..2..0..2....0..0..0..1....2..0..1..2....2..2..0..0....1..2..0..0
%e A206472 ..0..1..0..0....1..0..0..1....1..0..0..0....0..0..0..1....0..0..0..2
%e A206472 ..0..2..0..1....0..0..0..0....1..0..2..0....2..0..0..0....1..0..0..0
%e A206472 ..0..0..0..0....1..2..0..1....1..0..0..0....2..0..0..1....0..0..1..0
%e A206472 ..0..2..2..2....0..0..0..1....0..0..2..1....2..0..0..2....1..0..2..0
%Y A206472 Column 1 is A139818(n+3)
%K A206472 nonn,tabl,new
%O A206472 1,1
%A A206472 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206471
%S A206471 116281,82609921,4803737481,1457044540561,144660587865049,
%T A206471 30374762423377761,3848918441303655721,685519912635644947249,
%U A206471 97075380100800976220025,16048326634124121062046721
%N A206471 Number of (n+1)X8 0..2 arrays with every 2X2 subblock having zero permanent
%C A206471 Column 7 of A206472
%H A206471 R. H. Hardin, <a href="/A206471/b206471.txt">Table of n, a(n) for n = 1..210</a>
%F A206471 Empirical: a(n) = 187*a(n-1) +16890*a(n-2) -4251100*a(n-3) -40650128*a(n-4) +34942108176*a(n-5) -509473309952*a(n-6) -143301983645440*a(n-7) +3781912316451840*a(n-8) +331477180236791808*a(n-9) -11437248501456273408*a(n-10) -456953450334995742720*a(n-11) +19427954988478986715136*a(n-12) +378498296435438535049216*a(n-13) -20467094815444348981739520*a(n-14) -176626051842769499929444352*a(n-15) +13996085228807881237465137152*a(n-16) +30568383723039618710120693760*a(n-17) -6355164028117009141540627415040*a(n-18) +12457938945007485052056640684032*a(n-19) +1929828810876076957792938988929024*a(n-20) -9299483116586836848017963795611648*a(n-21) -389157136861521793604550385668718592*a(n-22) +2708325225472310296274920611243884544*a(n-23) +50924689860837777608863804409805537280*a(n-24) -447452547804564001571380763718647283712*a(n-25) -4125665523625190917601565450289790582784*a(n-26) +44428660564874781732775601609703285587968*a(n-27) +186837790537674954483787089018593126383616*a(n-28) -2618004719714992257591615094837678769700864*a(n-29) -3397132551485027226296690950170149841272832*a(n-30) +86176350300963117293366090863279754170073088*a(n-31) -38719009409154496417441531127358038313271296*a(n-32) -1390107790923631754160207042472622108971106304*a(n-33) +2057972867841973929472259334257306497261240320*a(n-34) +7785693287184378683612676651109988061281255424*a(n-35) -15061387791262419693934415229148488062591827968*a(n-36)
%e A206471 Some solutions for n=4
%e A206471 ..1..1..0..2..1..2..2..2....0..0..2..1..1..2..0..0....2..0..0..2..2..2..1..1
%e A206471 ..0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..2....0..0..0..0..0..0..0..0
%e A206471 ..1..0..2..1..2..1..1..1....0..0..2..1..0..1..0..2....0..2..0..1..0..0..0..2
%e A206471 ..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..2....0..2..0..0..0..2..0..0
%e A206471 ..2..0..1..2..0..2..1..1....0..2..1..0..2..1..0..2....0..0..0..2..0..0..0..1
%K A206471 nonn,new
%O A206471 1,1
%A A206471 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206470
%S A206470 29241,8963667,356190129,52552339515,3420616959081,377599682448675,
%T A206470 30374762423377761,2996776279527835851,262356222717008105625,
%U A206470 24807486299199737090355,2243325257562176033089041
%N A206470 Number of (n+1)X7 0..2 arrays with every 2X2 subblock having zero permanent
%C A206470 Column 6 of A206472
%H A206470 R. H. Hardin, <a href="/A206470/b206470.txt">Table of n, a(n) for n = 1..210</a>
%F A206470 Empirical: a(n) = 147*a(n-1) -783720*a(n-3) +22720320*a(n-4) +1208991744*a(n-5) -54468142848*a(n-6) -614613775104*a(n-7) +51672615856128*a(n-8) -95234427260928*a(n-9) -23998259218120704*a(n-10) +201605913386680320*a(n-11) +5689361505992048640*a(n-12) -76294817096637874176*a(n-13) -628017046749267886080*a(n-14) +13195491648030973624320*a(n-15) +14645283042051703701504*a(n-16) -1131405415550428045639680*a(n-17) +2665553144520224318423040*a(n-18) +45873154851254425563955200*a(n-19) -208791793059186748686336000*a(n-20) -694038525930408939356160000*a(n-21) +5010428057070448693739520000*a(n-22) -38126822008840601972244480000*a(n-24) +51697385774699121318297600000*a(n-25)
%e A206470 Some solutions for n=4
%e A206470 ..2..0..2..1..0..2..0....2..0..1..1..0..1..1....0..0..1..2..2..0..0
%e A206470 ..1..0..0..0..0..1..0....1..0..0..0..0..0..0....0..0..0..0..0..0..0
%e A206470 ..0..0..0..0..0..2..0....2..0..0..2..2..1..0....0..2..2..1..2..0..0
%e A206470 ..0..1..1..1..0..1..0....2..0..0..0..0..0..0....0..0..0..0..0..0..1
%e A206470 ..0..0..0..0..0..0..0....0..0..2..1..2..1..0....0..2..2..0..1..0..1
%K A206470 nonn,new
%O A206470 1,1
%A A206470 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206469
%S A206469 7225,958441,23707161,1693734025,61244865529,3420616959081,
%T A206469 144660587865049,7310031634630729,329657818912939449,
%U A206469 15990423345433902121,740325481079134598041,35313542504161253544969
%N A206469 Number of (n+1)X6 0..2 arrays with every 2X2 subblock having zero permanent
%C A206469 Column 5 of A206472
%H A206469 R. H. Hardin, <a href="/A206469/b206469.txt">Table of n, a(n) for n = 1..210</a>
%F A206469 Empirical: a(n) = 55*a(n-1) +978*a(n-2) -78848*a(n-3) +185192*a(n-4) +32350560*a(n-5) -254580608*a(n-6) -5080783360*a(n-7) +56605956096*a(n-8) +280495972352*a(n-9) -4555145707520*a(n-10) -1650631114752*a(n-11) +144197302616064*a(n-12) -230333727375360*a(n-13) -1370695862845440*a(n-14) +3206175906594816*a(n-15)
%e A206469 Some solutions for n=4
%e A206469 ..0..0..2..0..2..2....2..1..2..1..2..2....2..0..2..2..1..0....2..2..0..0..2..1
%e A206469 ..0..0..0..0..0..0....0..0..0..0..0..0....2..0..0..0..0..0....0..0..0..0..0..0
%e A206469 ..0..0..2..0..0..2....2..2..2..1..1..1....0..0..0..0..2..1....2..2..0..2..2..0
%e A206469 ..2..0..1..0..0..0....0..0..0..0..0..0....0..2..0..0..0..0....0..0..0..0..0..0
%e A206469 ..1..0..1..0..0..2....0..2..1..2..1..1....0..2..0..0..0..2....2..1..2..2..1..2
%K A206469 nonn,new
%O A206469 1,1
%A A206469 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206468
%S A206468 1849,106891,1896129,68352739,1693734025,52552339515,1457044540561,
%T A206468 43214597865811,1240051361495769,36294544907153707,
%U A206468 1052106476736321889,30674877583244503491,891844034416624515625,25973093752500918767899
%N A206468 Number of (n+1)X5 0..2 arrays with every 2X2 subblock having zero permanent
%C A206468 Column 4 of A206472
%H A206468 R. H. Hardin, <a href="/A206468/b206468.txt">Table of n, a(n) for n = 1..210</a>
%F A206468 Empirical: a(n) = 39*a(n-1) -11348*a(n-3) +76272*a(n-4) +429792*a(n-5) -3994496*a(n-6) +37171200*a(n-8) -43614208*a(n-9)
%e A206468 Some solutions for n=4
%e A206468 ..1..1..1..2..1....0..0..2..1..1....0..0..2..2..2....2..0..0..1..1
%e A206468 ..0..0..0..0..0....2..0..0..0..0....1..0..0..0..0....2..0..0..0..0
%e A206468 ..1..2..0..1..2....2..0..1..2..1....1..0..0..2..1....0..0..2..2..0
%e A206468 ..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
%e A206468 ..0..1..2..0..1....1..0..1..2..0....2..1..0..0..2....2..2..2..1..2
%K A206468 nonn,new
%O A206468 1,1
%A A206468 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206467
%S A206467 441,11025,110889,1896129,23707161,356190129,4803737481,69030731169,
%T A206467 955680252921,13523866505361,188928681187689,2659384141993089,
%U A206467 37267791691450329,523625523775949169,7345838128808607561
%N A206467 Number of (n+1)X4 0..2 arrays with every 2X2 subblock having zero permanent
%C A206467 Column 3 of A206472
%H A206467 R. H. Hardin, <a href="/A206467/b206467.txt">Table of n, a(n) for n = 1..210</a>
%F A206467 Empirical: a(n) = 15*a(n-1) +54*a(n-2) -1116*a(n-3) +1728*a(n-4) +10368*a(n-5) -20736*a(n-6)
%e A206467 Some solutions for n=4
%e A206467 ..2..2..2..1....2..2..0..0....0..1..0..1....1..2..0..0....0..0..2..0
%e A206467 ..0..0..0..0....0..0..0..1....0..2..0..2....0..0..0..2....0..0..2..0
%e A206467 ..1..2..1..0....2..0..0..0....0..1..0..1....1..0..0..0....0..0..0..0
%e A206467 ..0..0..0..0....2..0..0..1....0..1..0..1....0..0..1..0....2..2..1..0
%e A206467 ..2..1..2..1....2..0..0..2....0..1..0..1....1..0..2..0....0..0..0..0
%K A206467 nonn,new
%O A206467 1,1
%A A206467 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206466
%S A206466 121,1411,11025,106891,958441,8963667,82609921,767387611,7109019225,
%T A206466 65954564131,611588125681,5672733227307,52611971134921,
%U A206466 487976540781811,4525907460768225,41977494854894971,389337236836785721
%N A206466 Number of (n+1)X3 0..2 arrays with every 2X2 subblock having zero permanent
%C A206466 Column 2 of A206472
%H A206466 R. H. Hardin, <a href="/A206466/b206466.txt">Table of n, a(n) for n = 1..210</a>
%F A206466 Empirical: a(n) = 11*a(n-1) -176*a(n-3) +256*a(n-4)
%e A206466 Some solutions for n=4
%e A206466 ..2..2..1....2..1..0....2..2..1....0..0..0....0..0..1....0..0..0....2..0..1
%e A206466 ..0..0..0....0..0..0....0..0..0....1..2..0....1..0..1....0..1..2....2..0..1
%e A206466 ..0..0..2....1..1..0....2..1..1....0..0..0....2..0..2....0..0..0....1..0..1
%e A206466 ..2..0..1....0..0..0....0..0..0....1..1..2....0..0..1....2..1..2....0..0..1
%e A206466 ..0..0..2....0..0..2....1..1..0....0..0..0....1..0..0....0..0..0....2..0..1
%K A206466 nonn,new
%O A206466 1,1
%A A206466 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206465
%S A206465 25,1411,110889,68352739,61244865529,377599682448675,
%T A206465 3848918441303655721,232902549300945882026755,
%U A206465 26631802795003317433250214201,15822211869785299795988010016362883
%N A206465 Number of (n+1)X(n+1) 0..2 arrays with every 2X2 subblock having zero permanent
%C A206465 Diagonal of A206472
%H A206465 R. H. Hardin, <a href="/A206465/b206465.txt">Table of n, a(n) for n = 1..18</a>
%e A206465 Some solutions for n=4
%e A206465 ..0..0..0..2..0....2..1..0..2..1....1..2..1..0..1....1..2..2..2..1
%e A206465 ..0..1..0..1..0....0..0..0..0..0....0..0..0..0..1....0..0..0..0..0
%e A206465 ..0..0..0..1..0....1..0..0..2..0....2..1..2..0..1....0..1..0..0..2
%e A206465 ..0..2..0..0..0....0..0..0..0..0....0..0..0..0..2....0..0..0..0..1
%e A206465 ..0..1..0..0..0....1..2..0..1..2....1..2..2..0..1....1..1..2..0..2
%K A206465 nonn,new
%O A206465 1,1
%A A206465 R. H. Hardin (rhhardin(AT)att.net) Feb 08 2012

%I A206463
%S A206463 1,1,2,6,19,63,220,796,2951,11155,42846,166738,655990,2604868,
%T A206463 10426448,42023678,170407186,694723354,2845839124,11707587484,
%U A206463 48350311989,200377116719,833062172188,3473507707930,14521668486233,60859782366097,255639891601242
%N A206463  G.f. satisfies: A(x*D(-x)) = x where D(x) = g.f. of A014577, the dragon curve sequence.
%e A206463  G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 63*x^6 + 220*x^7 + 796*x^8 +...
%e A206463 such that A(x*D(-x)) = x and D(x) is the g.f. of the dragon curve:
%e A206463 D(x) = 1 + x + x^3 + x^4 + x^7 + x^8 + x^9 + x^12 + x^15 + x^16 + x^17 + x^19 + ...
%o A206463  (PARI) {A014577(n) = if( n%2, A014577(n\2), 1 - (n/2%2))}
%o A206463 {a(n)=local(DC=vector(n+1,k,(-1)^(k-1)*A014577(k-1)));polcoeff(serreverse(x*Ser(DC)),n)}
%o A206463 for(n=0,61,print1(a(n),","))
%Y A206463  Cf. A014577.
%K A206463 nonn,new
%O A206463 1,3
%A A206463 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206461
%S A206461 9,14849,1020000,48718765,2176455745,96888186864,4398044676096,
%T A206461 205891128374562,9999999993000000,505447028486892844,
%U A206461 26623333280864342016,1461920290375412323014,83668255425284748853824
%N A206461 Number of 0..n arrays of length n+7 avoiding the consecutive pattern 0..n
%C A206461 Subdiagonal 7 of A206455
%H A206461 R. H. Hardin, <a href="/A206461/b206461.txt">Table of n, a(n) for n = 1..210</a>
%F A206461 Empirical: a(n) = sum{i in 0..floor((n+7)/(n+1))} ((-1)^i*(n+1)^((n+7) -(n+1)*i)*binomial((n+7) -n*i,i))
%K A206461 nonn,new
%O A206461 1,1
%A A206461 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206460
%S A206460 8,5157,256012,9746876,362750400,13841186359,549755617280,
%T A206460 22876792100667,999999999400000,45949729862605855,2218611106738944000,
%U A206460 112455406951955165371,5976303958948911170240,332525673007965083334375
%N A206460 Number of 0..n arrays of length n+6 avoiding the consecutive pattern 0..n
%C A206460 Subdiagonal 6 of A206455
%H A206460 R. H. Hardin, <a href="/A206460/b206460.txt">Table of n, a(n) for n = 1..210</a>
%F A206460 Empirical: a(n) = sum{i in 0..floor((n+6)/(n+1))} ((-1)^i*(n+1)^((n+6) -(n+1)*i)*binomial((n+6) -n*i,i))
%K A206460 nonn,new
%O A206460 1,1
%A A206460 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206459
%S A206459 7,1791,64257,1950000,60459696,1977314738,68719456256,2541865795524,
%T A206459 99999999950000,4177248169342446,184884258894932736,
%U A206459 8650415919381195128,426878854210636550576,22168378200531005606250,1208925819614629174378496
%N A206459 Number of 0..n arrays of length n+5 avoiding the consecutive pattern 0..n
%C A206459 Subdiagonal 5 of A206455
%H A206459 R. H. Hardin, <a href="/A206459/b206459.txt">Table of n, a(n) for n = 1..210</a>
%F A206459 Empirical: a(n) = sum{i in 0..floor((n+5)/(n+1))} ((-1)^i*(n+1)^((n+5) -(n+1)*i)*binomial((n+5) -n*i,i))
%K A206459 nonn,new
%O A206459 1,1
%A A206459 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206458
%S A206458 6,622,16128,390125,10076832,282473877,8589932544,282429533565,
%T A206458 9999999996000,379749833577917,15407021574579456,665416609183171053,
%U A206458 30491346729331184928,1477891880035400377125,75557863725914323402752
%N A206458 Number of 0..n arrays of length n+4 avoiding the consecutive pattern 0..n
%C A206458 Subdiagonal 4 of A206455
%H A206458 R. H. Hardin, <a href="/A206458/b206458.txt">Table of n, a(n) for n = 1..210</a>
%F A206458 Empirical: a(n) = sum{i in 0..floor((n+4)/(n+1))} ((-1)^i*(n+1)^((n+4) -(n+1)*i)*binomial((n+4) -n*i,i))
%K A206458 nonn,new
%O A206458 1,1
%A A206458 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206457
%S A206457 5,216,4048,78050,1679508,40353460,1073741632,31381059366,
%T A206457 999999999700,34522712143568,1283918464548432,51185893014090250,
%U A206457 2177953337809370548,98526125335693358700,4722366482869645212928,239072435685151324846286
%N A206457 Number of 0..n arrays of length n+3 avoiding the consecutive pattern 0..n
%C A206457 Subdiagonal 3 of A206455
%H A206457 R. H. Hardin, <a href="/A206457/b206457.txt">Table of n, a(n) for n = 1..210</a>
%F A206457 Empirical: a(n) = sum{i in 0..floor((n+3)/(n+1))} ((-1)^i*(n+1)^((n+3) -(n+1)*i)*binomial((n+3) -n*i,i))
%K A206457 nonn,new
%O A206457 1,1
%A A206457 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206456
%S A206456 4,75,1016,15615,279924,5764787,134217712,3486784383,99999999980,
%T A206456 3138428376699,106993205379048,3937376385699263,155568095557812196,
%U A206456 6568408355712890595,295147905179352825824,14063084452067724990975
%N A206456 Number of 0..n arrays of length n+2 avoiding the consecutive pattern 0..n
%C A206456 Subdiagonal 2 of A206455
%H A206456 R. H. Hardin, <a href="/A206456/b206456.txt">Table of n, a(n) for n = 1..210</a>
%F A206456 Empirical: a(n) = sum{i in 0..floor((n+2)/(n+1))} ((-1)^i*(n+1)^((n+2) -(n+1)*i)*binomial((n+2) -n*i,i))
%K A206456 nonn,new
%O A206456 1,1
%A A206456 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206455
%S A206455 2,3,3,4,9,4,5,16,26,5,6,25,64,75,6,7,36,125,255,216,7,8,49,216,625,
%T A206455 1016,622,8,9,64,343,1296,3124,4048,1791,9,10,81,512,2401,7776,15615,
%U A206455 16128,5157,10,11,100,729,4096,16807,46655,78050,64257,14849,11,12,121,1000
%N A206455 T(n,k)=Number of 0..k arrays of length n avoiding the consecutive pattern 0..k
%C A206455 Table starts
%C A206455 .2....3.....4......5.......6.......7........8........9........10........11
%C A206455 .3....9....16.....25......36......49.......64.......81.......100.......121
%C A206455 .4...26....64....125.....216.....343......512......729......1000......1331
%C A206455 .5...75...255....625....1296....2401.....4096.....6561.....10000.....14641
%C A206455 .6..216..1016...3124....7776...16807....32768....59049....100000....161051
%C A206455 .7..622..4048..15615...46655..117649...262144...531441...1000000...1771561
%C A206455 .8.1791.16128..78050..279924..823542..2097152..4782969..10000000..19487171
%C A206455 .9.5157.64257.390125.1679508.5764787.16777215.43046721.100000000.214358881
%H A206455 R. H. Hardin, <a href="/A206455/b206455.txt">Table of n, a(n) for n = 1..10000</a>
%F A206455 Empirical: T(n,k) = sum{i=0..floor(n/(k+1))} ( (-1)^i * (k+1)^(n-(k+1)*i) * binomial(n-k*i,i) ) (after A076264)
%F A206455 Empirical for column k: a(n) = (k+1)*a(n-1) -a(n-(k+1))
%Y A206455 Column 2 is A076264
%Y A206455 Subdiagonal 1 is A048861(n+1)
%K A206455 nonn,tabl,new
%O A206455 1,1
%A A206455 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206454
%S A206454 8,64,512,4096,32768,262144,2097152,16777215,134217712,1073741632,
%T A206454 8589932544,68719456256,549755617280,4398044676096,35184355311616,
%U A206454 281474825715713,2251798471507992,18014386698322304,144115084996645888
%N A206454 Number of 0..7 arrays of length n avoiding the consecutive pattern 0..7
%C A206454 Column 7 of A206455
%H A206454 R. H. Hardin, <a href="/A206454/b206454.txt">Table of n, a(n) for n = 1..210</a>
%F A206454 Empirical: a(n) = 8*a(n-1) -a(n-8)
%F A206454 Empirical: a(n) = sum{i in 0..floor(n/8)} ((-1)^i*8^(n-8*i)*binomial(n-7*i,i))
%K A206454 nonn,new
%O A206454 1,1
%A A206454 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206453
%S A206453 7,49,343,2401,16807,117649,823542,5764787,40353460,282473877,
%T A206453 1977314738,13841186359,96888186864,678216484506,4747509626755,
%U A206453 33232527033825,232627406762898,1628389870025548,11398715248992477,79790909854760475
%N A206453 Number of 0..6 arrays of length n avoiding the consecutive pattern 0..6
%C A206453 Column 6 of A206455
%H A206453 R. H. Hardin, <a href="/A206453/b206453.txt">Table of n, a(n) for n = 1..210</a>
%F A206453 Empirical: a(n) = 7*a(n-1) -a(n-7)
%F A206453 Empirical: a(n) = sum{i in 0..floor(n/7)} ((-1)^i*7^(n-7*i)*binomial(n-6*i,i))
%K A206453 nonn,new
%O A206453 1,1
%A A206453 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206452
%S A206452 6,36,216,1296,7776,46655,279924,1679508,10076832,60459696,362750400,
%T A206452 2176455745,13058454546,78349047768,470084209776,2820444798960,
%U A206452 16922306043360,101531659804415,609176900371944,3654983053183896
%N A206452 Number of 0..5 arrays of length n avoiding the consecutive pattern 0..5
%C A206452 Column 5 of A206455
%H A206452 R. H. Hardin, <a href="/A206452/b206452.txt">Table of n, a(n) for n = 1..210</a>
%F A206452 Empirical: a(n) = 6*a(n-1) -a(n-6)
%F A206452 Empirical: a(n) = sum{i in 0..floor(n/6)} ((-1)^i*6^(n-6*i)*binomial(n-5*i,i))
%K A206452 nonn,new
%O A206452 1,1
%A A206452 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206451
%S A206451 5,25,125,625,3124,15615,78050,390125,1950000,9746876,48718765,
%T A206451 243515775,1217188750,6083993750,30410221874,152002390605,
%U A206451 759768437250,3797624997500,18982040993750,94879794746876,474246971343775
%N A206451 Number of 0..4 arrays of length n avoiding the consecutive pattern 0..4
%C A206451 Column 4 of A206455
%H A206451 R. H. Hardin, <a href="/A206451/b206451.txt">Table of n, a(n) for n = 1..210</a>
%F A206451 Empirical: a(n) = 5*a(n-1) -a(n-5)
%F A206451 Empirical: a(n) = sum{i in 0..floor(n/5)} ((-1)^i*5^(n-5*i)*binomial(n-4*i,i))
%K A206451 nonn,new
%O A206451 1,1
%A A206451 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206450
%S A206450 4,16,64,255,1016,4048,16128,64257,256012,1020000,4063872,16191231,
%T A206450 64508912,257015648,1023998720,4079803649,16254705684,64761807088,
%U A206450 258023229632,1028013114879,4095797753832,16318429208240,65015693603328
%N A206450 Number of 0..3 arrays of length n avoiding the consecutive pattern 0..3
%C A206450 Column 3 of A206455
%H A206450 R. H. Hardin, <a href="/A206450/b206450.txt">Table of n, a(n) for n = 1..210</a>
%F A206450 Empirical: a(n) = 4*a(n-1) -a(n-4)
%F A206450 Empirical: a(n) = sum{i in 0..floor(n/4)} ((-1)^i*4^(n-4*i)*binomial(n-3*i,i))
%K A206450 nonn,new
%O A206450 1,1
%A A206450 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206333
%S A206333 3,7,251,61223
%N A206333 Smallest prime q such that, starting with q, there are prime(n)-1 consecutive primes = {1..prime(n)-1} modulo prime(n).
%e A206333 Let p(n) = prime(n), then
%e A206333 a(1)=3 because 3 is smallest prime = ( 1 modulo p(1)) = 1 mod 2;
%e A206333 a(2)=7 because 2 smallest consecutive primes {7,11}= {1,2} modulo p(2) = {1,2} mod 3;
%e A206333 a(3) = 251 because {251,257,263,269} = {1,2,3,4} modulo p(3)= {1,2,3,4} mod 5;
%e A206333 a(4)= 61223 because {61223,61231,61253,61261,61283,61291} = {1,2,3,4,5,6} modulo p(4)= {1,2,3,4,5,6} mod 7.
%t A206333 Table[n = 1; While[Mod[Prime[Range[n, n+p-2]], p] != Range[p-1], n++]; Prime[n], {p, Prime[Range[4]]}] (* T. D. Noe, Feb 07 2012 *)
%K A206333 hard,more,nonn,new
%O A206333 1,1
%A A206333 Moshe Levin (moshe.levin(AT)mail.ru), Feb 06 2012

%I A206426
%S A206426 0,19,60,84,115,184,231,279,400,483,580,799,931,1104,1495,1764,2040,
%T A206426 2739,3220,3783,5056,5824,6831,9108,10675,12283,16356,19159,22440,
%U A206426 29859,34335,40204,53475,62608,71980,95719,112056,131179,174420,200508,234715,312064
%N A206426 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+161)^2 = y^2.
%t A206426 LinearRecurrence[{1,0,0,0,0,0,0,0,6,-6,0,0,0,0,0,0,0,-1,1}, {0,19,60,84,115,184,231,279,400,483,580,799,931,1104,1495,1764,2040,2739,3220}, 120]
%Y A206426 Cf. A129544, A129837, A129992
%K A206426 nonn,new
%O A206426 1,2
%A A206426 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A206257
%S A206257 14,98,2702,524174,940898,101687054,9034502498,19726764302,
%T A206257 3826890587534,86749292044898,742397047217294,144021200269567502,
%U A206257 832966693180608098,27939370455248878094,5420093847118012782734,7998146101170906912098,1051470266970439230972302
%N A206257 Values of S(1) such that any Mersenne prime with an odd exponent p divides S(p-2), where S(n) == S(n-1)^2 - 2 (mod M(p)).
%H A206257 Arkadiusz Wesolowski, <a href="/A206257/b206257.txt">Table of n, a(n) for n = 1..150</a>
%H A206257 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas-Lehmer_primality_test">Lucas-Lehmer primality test</a>
%F A206257 Union of sequences a(0) = 14, a(1) = 2702; a(n) = 194*a(n-1) - a(n-2) and b(0) = 98, b(1) = 940898; b(n) = 9602*b(n-1) - b(n-2).
%F A206257 a(n) = A018844(n)^2 - 2.
%t A206257 nn = 17; t1 = LinearRecurrence[{194, -1}, {14, 2702}, nn]; t2 = LinearRecurrence[{9602, -1}, {98, 940898}, nn]; t3 = Select[t2, # < t1[[-1]]&]; Union[t1, t3]
%K A206257 easy,nonn,new
%O A206257 1,1
%A A206257 Arkadiusz Wesolowski (wesolowski(AT)aol.pl), Feb 05 2012

%I A206351
%S A206351 1,3,16,105,715,4896,33553,229971,1576240,10803705,74049691,507544128,
%T A206351 3478759201,23843770275,163427632720,1120149658761,7677619978603,
%U A206351 52623190191456,360684711361585,2472169789339635
%N A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.
%C A206351 A Pell sequence related to Heronian triangles (rational triangles), see A206334. The connection is this: consider the problem of finding triangles with area a positive integer n, and with sides (a, b, n) where a, b are rational. Note that n is both the area and one side. For many values of n this is not possible, and the sequence of such numbers, n, is quite erratic (see A206334).  Nonetheless, each term in this sequence is such a value of n. For example, for n = 105 you can take the other two sides, a and b, to be 10817/104, and 233/104 and the area will equal n, i.e. 105.
%H A206351 Reinhard Zumkeller, <a href="/A206351/b206351.txt">Table of n, a(n) for n = 1..1000</a>
%H A206351 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A206351 a(n) = 4/5+1/10*((7/2-3/2*sqrt(5))^(n-1)+(7/2+3/2*sqrt(5))^(n-1))+1/10*sqrt(5)*((7/2 +3/2*sqrt(5))^(n-1)-(7/2-3/2*sqrt(5))^(n-1)). - Paolo P. Lava, Feb 07 2012.
%F A206351 G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3). a(n) = A081018(n-1)+1. - Bruno Berselli, Feb 07 2012
%p A206351 genZ := proc(n)
%p A206351 local start;
%p A206351 option remember;
%p A206351     start := [1, 3];
%p A206351     if n < 3 then start[n]
%p A206351     else 7*genZ(n - 1) - genZ(n - 2) - 4
%p A206351     end if
%p A206351 end proc:
%p A206351 seq(genZ(n),n=1..20);
%t A206351 LinearRecurrence[{8,-8,1},{1,3,16},50] (* Charles R Greathouse IV, Feb 07 2012 *)
%t A206351 RecurrenceTable[{a[1] == 1, a[2] == 3, a[n] == 7 a[n - 1] - a[n - 2] - 4}, a, {n, 20}] (* Bruno Berselli, Feb 07, 2012 *)
%o A206351 (PARI) Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 07 2012
%o A206351 (Haskell)
%o A206351 a206351 n = a206351_list !! (n-1)
%o A206351 a206351_list = 1 : 3 : map (subtract 4)
%o A206351                (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)
%o A206351 -- Reinhard Zumkeller, Feb 08 2012
%Y A206351 Subsequence of A206334.
%K A206351 nonn,easy,new
%O A206351 1,2
%A A206351 James R. Buddenhagen (jbuddenh(AT)gmail.com), Feb 06 2012

%I A205947
%S A205947 561,2465,62745,162401,656601,1909001,5444489,11921001,19384289,
%T A205947 26719701,45318561,84350561,151530401,174352641,221884001,230996949,
%U A205947 275283401,434932961,662086041,684106401,689880801,710382401
%N A205947 Carmichael numbers not congruent to 1 modulo 6.
%t A205947 Select[Range[100000], !PrimeQ[#] && IntegerQ[(#-1)/CarmichaelLambda[#]] && !Mod[#,6]==1&]
%Y A205947 Cf. A002997.
%K A205947 nonn,new
%O A205947 1,1
%A A205947 José María Grau Ribas (grau.ribas(AT)gmail.com), Feb 02 2012

%I A199105
%S A199105 24,48,56,72,80,88,96,112,144,152,160,168,176,184,192,208,216,224,240,
%T A199105 248,264,288,304,320,336,344,352,368,376,384,392,400,416,432,448,456,
%U A199105 464,472,480,496,504,528,536,552,560,568,576,592,608,616,624
%N A199105 Numbers n such that lambda(n) < A011773(n) < phi(n), where lambda is the Carmichael reduced totient function and phi the Euler totien function.
%C A199105 A002322(n) divides A011773(n) and A011773(n) divides A000010(n).
%t A199105 A011773[p_,s_] := (p-1)*p^(s-1); A011773[n_] := {aux=1;Do[aux=LCM[aux,A011773[FactorInteger[n][[i,1]], FactorInteger[n][[i,2]]]], {i,Length[FactorInteger[n]]}]; aux}[[1]]; Select[Range[1000], CarmichaelLambda[#] < A011773[#] < EulerPhi[#]&]
%Y A199105 Cf. A002322, A011773, A000010.
%K A199105 nonn,new
%O A199105 1,1
%A A199105 José María Grau Ribas (grau.ribas(AT)gmail.com), Jan 31 2012

%I A206349
%S A206349 380,506,3796,6006,8976,9186,10920,12896,14476,14800,15386,32326,
%T A206349 38460,39536,40420,41456,43430,60076,74676,76986,82530,87390,99486,
%U A206349 107926,112840,126996,127920,144326,179566,181986,188526,193006,194616,205200,217520,230370
%N A206349 Even numbers n such that 6n+1, 12n+1, 18n+1, 36n+1 and 72n+1 are all primes.
%C A206349 (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) is a Carmichael number for all n in this sequence.
%D A206349 Jack Chernick, On Fermat's simple theorem, Bull. Amer. Math. Soc., Volume 45, Number 4 (1939), 269-274.
%t A206349 Select[Range[250000], PrimeQ[6 #+1] && PrimeQ[12 #+1] && PrimeQ[18 #+1] && PrimeQ[36 #+1] && PrimeQ[72 #+1] && Mod[#,2] == 0&]
%K A206349 nonn,new
%O A206349 1,1
%A A206349 José María Grau Ribas (grau.ribas(AT)gmail.com), Feb 06 2012

%I A206423
%S A206423 12,7,19,26,45,71,116,187,303,490,793,1283,2076,3359,5435,8794,14229,
%T A206423 23023,37252,60275,97527,157802,255329,413131,668460,1081591,1750051,
%U A206423 2831642,4581693,7413335,11995028,19408363,31403391,50811754,82215145,133026899
%N A206423 Fibonacci sequence beginning 12 7.
%t A206423 LinearRecurrence[{1,1},{12,7},60]
%Y A206423 Cf. A000045
%K A206423 nonn,new
%O A206423 1,1
%A A206423 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A206422
%S A206422 11,9,20,29,49,78,127,205,332,537,869,1406,2275,3681,5956,9637,15593,
%T A206422 25230,40823,66053,106876,172929,279805,452734,732539,1185273,1917812,
%U A206422 3103085,5020897,8123982,13144879,21268861,34413740,55682601,90096341,145778942
%N A206422 Fibonacci sequence beginning 11 9.
%t A206422 LinearRecurrence[{1, 1}, {11, 9}, 60]
%Y A206422 Cf. A000045
%K A206422 nonn,new
%O A206422 1,1
%A A206422 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A206420
%S A206420 11,8,19,27,46,73,119,192,311,503,814,1317,2131,3448,5579,9027,14606,
%T A206420 23633,38239,61872,100111,161983,262094,424077,686171,1110248,1796419,
%U A206420 2906667,4703086,7609753,12312839,19922592,32235431,52158023,84393454,136551477
%N A206420 Fibonacci sequence beginning 11 8.
%t A206420 LinearRecurrence[{1, 1}, {11, 8}, 60]
%Y A206420 Cf. A000045
%K A206420 nonn,new
%O A206420 1,1
%A A206420 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A206419
%S A206419 11,7,18,25,43,68,111,179,290,469,759,1228,1987,3215,5202,8417,13619,
%T A206419 22036,35655,57691,93346,151037,244383,395420,639803,1035223,1675026,
%U A206419 2710249,4385275,7095524,11480799,18576323,30057122,48633445,78690567,127324012
%N A206419 Fibonacci sequence beginning 11 7.
%t A206419 LinearRecurrence[{1, 1}, {11, 7}, 60]
%Y A206419 Cf. A000045
%K A206419 nonn,new
%O A206419 1,1
%A A206419 Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Feb 07 2012

%I A206397
%S A206397 1,1,3,21,171,1821,24123,373941,6693291,135897741,3081969243,
%T A206397 77250233061,2120715880011,63277499072061,2039050439495163,
%U A206397 70571948084252181,2610905715855178731,102824333281385113581
%N A206397 E.g.f. A(x) = series reversion of log(1+x)-x^3/3.
%F A206397 a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, (-1)^j/(k-j)!*sum(i=0..min(j,(n+j-1)/3),(1/i!)*(-1)^i* stirling1(n-3*i+j-1,j-i)/(3^i*(n-3*i+j-1)!))))), n>0.
%o A206397 (Maxima) a(n):=(sum((n+k-1)!*sum((-1)^j/(k-j)!*sum((1/i!)*(-1)^i*stirling1(n-3*i+j-1,j-i)/(3^i*(n-3*i+j-1)!),i,0,min(j,(n+j-1)/3)),j,0,k),k,0,n-1));
%K A206397 nonn,new
%O A206397 1,3
%A A206397 Vladimir Kruchinin (kru(AT)ie.tusur.ru), Feb 07 2012

%I A206416
%S A206416 4,5,6,8,10,12,14,16,18,20,22,24,24,26,28,29,30,32,33,34,36,37,38,40
%N A206416 Achromatic number of K_4 X K_n.
%D A206416 Hornak, Mirko and Puntigan, Jozef, On the achromatic number of K_m X K_n. In Graphs and other combinatorial topics (Prague, 1982), 118-123, Teubner-Texte Math., 59, Teubner, Leipzig, 1983.
%D A206416 Hornak, Mirko and Pcola, Stefan, Achromatic number of K_5 X K_n for small n. Czechoslovak Math. J. 53 (128) (2003), no. 4, 963-988.
%D A206416 Hornak, Mirko and Pcola, Stefan, Achromatic number of K_5 X K_n for large n. Discrete Math. 234 (2001), no. 1-3, 159-169.
%F A206416 For n >= 25, a(n) = floor(5n/3).
%Y A206416 Cf. A206415.
%K A206416 nonn,more,new
%O A206416 1,1
%A A206416 N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2012

%I A206304
%S A206304 1,2,6,22,80,128,2940,63072,932088,11628648,114829968,417677856,
%T A206304 21173151792,869103962400,23125766258208,492858262277472,
%U A206304 7960636847682816,46152793911618432,3484964629275707328,212667378331722666240
%V A206304 1,2,6,22,80,128,-2940,-63072,-932088,-11628648,-114829968,-417677856,
%W A206304 21173151792,869103962400,23125766258208,492858262277472,
%X A206304 7960636847682816,46152793911618432,-3484964629275707328,-212667378331722666240
%N A206304 E.g.f. is series reversion of x-log(1+x)^2
%F A206304 a(n)= ((n-1)!*sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, binomial(k,j)*sum(i=0..j, ((-1)^(i)*binomial(j,i)*(2*(j-i))!*stirling1(n+j-i-1,2*(j-i)))/(n+j-i-1)!)))), n>0.
%o A206304 (Maxima) a(n):= ((n-1)!*sum(binomial(n+k-1,n-1)*sum(binomial(k,j)*sum(((-1)^(i)*binomial(j,i)*(2*(j-i))!*stirling1(n+j-i-1,2*(j-i)))/(n+j-i-1)!,i,0,j),j,0,k),k,0,n-1));
%K A206304 sign,new
%O A206304 1,2
%A A206304 Vladimir Kruchinin (kru(AT)ie.tusur.ru), Feb 06 2012

%I A206415
%S A206415 5,6,7,10,12,15,16,18,19,22,23,25,27,29,30,32,34,36,38,40,42,44,46,48,
%T A206415 49,50,52,54,55,57,58,60,61,63,64,66,67,69,71,72,74,76
%N A206415 Achromatic number of K_5 X K_n.
%D A206415 Hornak, Mirko and Pcola, Stefan, Achromatic number of K_5 X K_n for small n. Czechoslovak Math. J. 53 (128) (2003), no. 4, 963-988.
%D A206415 Hornak, Mirko and Pcola, Stefan, Achromatic number of K_5 X K_n for large n. Discrete Math. 234 (2001), no. 1-3, 159-169.
%F A206415 For n >= 43, a(n) = floor(9n/5).
%Y A206415 Cf. A206416.
%K A206415 nonn,more,new
%O A206415 1,1
%A A206415 N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2012

%I A206402
%S A206402 1,3,26,422,9684,284536,10205264,432507008,21149344320,1172055816864,
%T A206402 72593488746624,4969455399927168,372585629959484928,
%U A206402 30363657581135890176,2672420848359072517632,252632488649577779398656,25529176319428221234402816,2746226455049097879478060032
%N A206402 E.g.f. A(x) satisfies: exp(A(x)) = x + exp(2*A(x)^2), with A(0) = 0.
%F A206402 E.g.f.: A(x) = Series_Reversion( exp(x) - exp(2*x^2) ).
%e A206402 E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 422*x^4/4! + 9684*x^5/5! +...
%e A206402 where A( exp(x) - exp(2*x^2) ) = x.
%e A206402 Related expansions:
%e A206402 exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...
%e A206402 exp(2*A(x)^2) = 1 + 4*x^2/2! + 36*x^3/3! + 572*x^4/4! + 13000*x^5/5! +...
%o A206402 (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(X)-exp(2*X^2)), n))}
%o A206402 for(n=0,30,print1(a(n),", "))
%Y A206402 Cf. A138014, A206401, A206403, A206404, A206405.
%K A206402 nonn,new
%O A206402 1,2
%A A206402 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206401
%S A206401 1,2,11,126,2049,42012,1047507,30867540,1049597685,40441973328,
%T A206401 1741357779039,82865509846776,4318613855629605,244629863660429712,
%U A206401 14965278826983897303,983295107764013223504,69061868853286423944249,5163430410995824208371968
%N A206401 E.g.f. A(x) satisfies: exp(A(x)) = x + exp(3*A(x)^2/2), with A(0) = 0.
%F A206401 E.g.f.: A(x) = Series_Reversion( exp(x) - exp(3*x^2/2) ).
%e A206401 E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 126*x^4/4! + 2049*x^5/5! +...
%e A206401 where A( exp(x) - exp(3*x^2/2) ) = x.
%e A206401 Related expansions:
%e A206401 exp(A(x)) = 1 + x + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
%e A206401 exp(3*A(x)^2/2) = 1 + 3*x^2/2! + 18*x^3/3! + 195*x^4/4! + 3090*x^5/5! +...
%o A206401 (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(X)-exp(3*X^2/2)), n))}
%o A206401 for(n=0,30,print1(a(n),", "))
%Y A206401 Cf. A138014, A206402, A206403, A206404, A206405.
%K A206401 nonn,new
%O A206401 1,2
%A A206401 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206404
%S A206404 1,3,26,364,7074,176108,5348132,191725840,7924856460,371061933552,
%T A206404 19411323110904,1122067341369984,71024428188382200,
%U A206404 4885895673623299008,362955565203398550768,28957167717593649778176,2469386593299982674585744,224154175905500071395278592
%N A206404 E.g.f. A(x) satisfies: exp(-A(x)) = exp(-A(x)^2) - x, with A(0) = 0.
%F A206404 E.g.f.: A(x) = Series_Reversion( exp(-x^2) - exp(-x) ).
%e A206404 E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 364*x^4/4! + 7074*x^5/5! +...
%e A206404 where A( exp(-x^2) - exp(-x) ) = x.
%e A206404 Related expansions:
%e A206404 exp(-A(x)) = 1 - x - 2*x^2/2! - 18*x^3/3! - 250*x^4/4! - 4840*x^5/5! +...
%e A206404 exp(-A(x)^2) = 1 - 2*x^2/2! - 18*x^3/3! - 250*x^4/4! - 4840*x^5/5! +...
%o A206404 (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(-X^2) - exp(-X)), n))}
%o A206404 for(n=0,30,print1(a(n),", "))
%Y A206404 Cf. A138014, A206401, A206402, A206403, A206405.
%K A206404 nonn,new
%O A206404 1,2
%A A206404 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206403
%S A206403 1,3,26,398,8604,239296,8135504,326921192,15159790680,796766681184,
%T A206403 46805302872624,3039065898588144,216125148650657232,
%U A206403 16706734205424667296,1394789126514873632832,125073511937467759505760,11989203887017099078716384,1223407961244225521367780096
%N A206403  E.g.f. A(x) satisfies: exp(A(x)) = 2*exp(A(x)^2) - (1-x), with A(0) = 0.
%F A206403  E.g.f.: A(x) = Series_Reversion( 1 + exp(x) - 2*exp(x^2) ).
%e A206403  E.g.f.: A(x) = x + 3*x^2/2! + 26*x^3/3! + 398*x^4/4! + 8604*x^5/5! +...
%e A206403 where A( 1 + exp(x) - 2*exp(x^2) ) = x.
%e A206403 Related expansions:
%e A206403 exp(A(x)) = 1 + x + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...
%e A206403 2*exp(A(x)^2) = 2 + 4*x^2/2! + 36*x^3/3! + 548*x^4/4! + 11800*x^5/5! +...
%o A206403  (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(1+exp(X)-2*exp(X^2)), n))}
%o A206403 for(n=0,30,print1(a(n),", "))
%Y A206403  Cf. A138014, A206401, A206402, A206404, A206405.
%K A206403 nonn,new
%O A206403 1,2
%A A206403 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206405
%S A206405 1,3,20,218,3414,70306,1789850,54071216,1886496960,74588759664,
%T A206405 3295393803888,160898970043632,8603780292835896,500078481148348176,
%U A206405 31391957137745933088,2116613399519305596384,152558384742741641353056,11705479592386152200155200
%N A206405  E.g.f. A(x) satisfies: exp(A(x)) = x + 2*exp(A(x)^2) - exp(A(x)^3), with A(0) = 0.
%F A206405  E.g.f.: A(x) = Series_Reversion( exp(x) - 2*exp(x^2) + exp(x^3) ).
%e A206405  E.g.f.: A(x) = x + 3*x^2/2! + 20*x^3/3! + 218*x^4/4! + 3414*x^5/5! +...
%e A206405 where A( exp(x) - 2*exp(x^2) + exp(x^3) ) = x.
%e A206405 Related expansions:
%e A206405 exp(A(x)) = 1 + x + 4*x^2/2! + 30*x^3/3! + 344*x^4/4! + 5470*x^5/5! +...
%e A206405 2*exp(A(x)^2) = 2 + 4*x^2/2! + 36*x^3/3! + 452*x^4/4! + 7480*x^5/5! +...
%e A206405 exp(A(x)^3) = 1 + 6*x^3/3! + 108*x^4/4! + 2010*x^5/5! +...
%o A206405  (PARI) {a(n)=local(X=x+x*O(x^n));if(n<1, 0, n!*polcoeff(serreverse(exp(X)-2*exp(X^2)+exp(X^3)), n))}
%o A206405 for(n=0,30,print1(a(n),", "))
%Y A206405  Cf. A138014, A206401, A206402, A206403, A206404.
%K A206405 nonn,new
%O A206405 1,2
%A A206405 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 07 2012

%I A206414
%S A206414 81,423,423,2457,5199,2457,15087,69585,69585,15087,94761,953865,
%T A206414 2073999,953865,94761,600519,13148043,62351943,62351943,13148043,
%U A206414 600519,3818649,181467207,1876995285,4099250889,1876995285,181467207,3818649,24314127
%N A206414 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206414 Table starts
%C A206414 .......81.........423...........2457.............15087.................94761
%C A206414 ......423........5199..........69585............953865..............13148043
%C A206414 .....2457.......69585........2073999..........62351943............1876995285
%C A206414 ....15087......953865.......62351943........4099250889..........269845754715
%C A206414 ....94761....13148043.....1876995285......269845754715........38883964087761
%C A206414 ...600519...181467207....56515649709....17769571483479......5607670598448153
%C A206414 ..3818649..2505335985..1701726564411..1170258128101905....808953988411331487
%C A206414 .24314127.34591087977.51240524315871.77072446958972853.116711118064834566675
%H A206414 R. H. Hardin, <a href="/A206414/b206414.txt">Table of n, a(n) for n = 1..143</a>
%e A206414 Some solutions for n=4 k=3
%e A206414 ..1..2..0..2....0..1..2..0....2..1..1..1....0..1..2..0....0..1..1..1
%e A206414 ..1..1..2..2....1..0..1..2....2..1..1..2....1..2..2..0....0..2..0..2
%e A206414 ..1..2..2..1....1..1..2..0....1..1..1..1....0..1..1..2....0..1..0..1
%e A206414 ..2..0..2..2....0..1..2..2....1..2..2..1....0..1..2..1....2..1..2..1
%e A206414 ..2..0..0..2....2..0..1..1....2..2..1..0....2..0..1..0....2..0..0..0
%K A206414 nonn,tabl,new
%O A206414 1,1
%A A206414 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206413
%S A206413 3818649,2505335985,1701726564411,1170258128101905,808953988411331487,
%T A206413 560493388205613135219,388756746462491144596125,
%U A206413 269773510833311684838942831,187249435580728308774605385177
%N A206413 Number of (n+1)X8 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206413 Column 7 of A206414
%H A206413 R. H. Hardin, <a href="/A206413/b206413.txt">Table of n, a(n) for n = 1..12</a>
%e A206413 Some solutions for n=4
%e A206413 ..2..1..0..2..2..0..1..0....2..0..1..0..0..0..2..1....1..0..2..1..1..0..2..1
%e A206413 ..0..2..1..0..0..1..2..1....0..0..0..0..0..0..2..1....0..1..0..2..1..0..2..1
%e A206413 ..0..2..1..1..0..1..1..1....0..1..1..1..0..1..0..2....0..0..0..2..1..0..2..2
%e A206413 ..1..0..2..2..1..1..1..1....1..0..1..1..0..0..2..0....0..2..2..0..2..1..0..2
%e A206413 ..0..2..1..1..0..0..0..1....0..0..0..1..1..0..0..1....0..2..0..2..0..2..1..0
%K A206413 nonn,new
%O A206413 1,1
%A A206413 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206412
%S A206412 600519,181467207,56515649709,17769571483479,5607670598448153,
%T A206412 1772313933215113017,560493388205613135219,177302245070698531560585,
%U A206412 56092636774114046549483283,17746700710940391048253081503
%N A206412 Number of (n+1)X7 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206412 Column 6 of A206414
%H A206412 R. H. Hardin, <a href="/A206412/b206412.txt">Table of n, a(n) for n = 1..44</a>
%e A206412 Some solutions for n=4
%e A206412 ..2..1..0..2..1..2..2....0..2..1..0..1..2..0....0..2..1..2..0..1..2
%e A206412 ..1..1..0..2..1..1..1....2..0..2..1..2..0..2....0..2..2..0..2..0..1
%e A206412 ..2..1..0..2..1..0..0....2..0..2..2..0..2..1....1..0..2..2..1..2..0
%e A206412 ..1..2..1..0..2..1..0....0..2..1..2..2..2..2....0..2..1..1..1..2..2
%e A206412 ..0..1..1..0..0..2..1....1..0..2..2..2..2..0....1..0..2..1..1..1..1
%K A206412 nonn,new
%O A206412 1,1
%A A206412 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206411
%S A206411 94761,13148043,1876995285,269845754715,38883964087761,
%T A206411 5607670598448153,808953988411331487,116711118064834566675,
%U A206411 16839058163900510312613,2429571141623141719964529,350544958031922890718492975
%N A206411 Number of (n+1)X6 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206411 Column 5 of A206414
%H A206411 R. H. Hardin, <a href="/A206411/b206411.txt">Table of n, a(n) for n = 1..210</a>
%F A206411 Empirical: a(n) = 326*a(n-1) -40232*a(n-2) +2625135*a(n-3) -102664380*a(n-4) +2483992594*a(n-5) -34054261073*a(n-6) +103742898710*a(n-7) +5425256115513*a(n-8) -108748874196029*a(n-9) +804683710872587*a(n-10) +2446678419327350*a(n-11) -105763975134186403*a(n-12) +881252218606156059*a(n-13) -995930084566858402*a(n-14) -41089452399216500107*a(n-15) +354301862067296697562*a(n-16) -786316291223027549741*a(n-17) -6833495162564130297562*a(n-18) +58894027033833075549821*a(n-19) -126174717794483840139905*a(n-20) -646419224150759600616616*a(n-21) +4858118678559133567903611*a(n-22) -7937915943690187608650707*a(n-23) -39018859939361196862635245*a(n-24) +216243213933651127286563249*a(n-25) -211108025777725211955307225*a(n-26) -1451724837983321979560422264*a(n-27) +5326972804948486086071230574*a(n-28) -1553776221107177921870106372*a(n-29) -31628411893576285291975387285*a(n-30) +72451639942200250029607652155*a(n-31) +30437559570605803869248014809*a(n-32) -396205187667773917409624594067*a(n-33) +528745042199175081528405237379*a(n-34) +690922961048740174580726697966*a(n-35) -2857385678388258875963047326199*a(n-36) +1853983705732808906593921494594*a(n-37) +5631327428566960787847827653286*a(n-38) -11926837565080346359469292396177*a(n-39) +1306634704432038839386265213304*a(n-40) +23760492003556783103212164780910*a(n-41) -28773237046156020272258832541327*a(n-42) -11139732502642525897601218183651*a(n-43) +57044915187466168684882488318548*a(n-44) -39301873629215708342907550983245*a(n-45) -38346144352222006815102242400887*a(n-46) +81250880311335856546165574959573*a(n-47) -28215415604332959931456606864272*a(n-48) -56724884111560914353370204260817*a(n-49) +70166815167995640081205601978139*a(n-50) -7516611640501782949119740142544*a(n-51) -45867015225519496282772970999220*a(n-52) +37093155885480928803041333007577*a(n-53) +2463345273627342113172168257660*a(n-54) -21475456406439655375685209518924*a(n-55) +11954018013055489620711427131837*a(n-56) +2299785430994497692988229377550*a(n-57) -5875877499685008598684447479201*a(n-58) +2280011629545083635037121735457*a(n-59) +672255618582305423900990337036*a(n-60) -928012582168332927633475694274*a(n-61) +237859929856085748014898299278*a(n-62) +102909452202118791193929516721*a(n-63) -83482695914804464255741959715*a(n-64) +11481398938406435255695989428*a(n-65) +8829802594826302806896776065*a(n-66) -4124483213111295097407111355*a(n-67) +74512575374728286264782785*a(n-68) +413647780649841499350298486*a(n-69) -101081218157260142111005917*a(n-70) -13060806063257772877571391*a(n-71) +9845861207796847640927530*a(n-72) -853704833929096428290937*a(n-73) -425800621353993002646301*a(n-74) +102452645011725971110187*a(n-75) +5642589248948002835528*a(n-76) -4532595847904168190307*a(n-77) +268647036281784693325*a(n-78) +104769732997178224719*a(n-79) -14620868331429200758*a(n-80) -1163575402960936720*a(n-81) +317239605542649458*a(n-82) +912000032134398*a(n-83) -3645417691373520*a(n-84) +113607271814048*a(n-85) +21783470035840*a(n-86) -1026098651472*a(n-87) -55400600736*a(n-88) +2760604544*a(n-89) +23077376*a(n-90)
%e A206411 Some solutions for n=4
%e A206411 ..1..1..1..0..2..1....0..0..0..2..1..2....1..0..2..1..2..1....2..1..2..2..2..0
%e A206411 ..2..1..1..1..0..2....1..0..1..0..2..1....2..1..0..2..2..2....2..1..1..1..1..2
%e A206411 ..0..2..2..1..1..0....0..1..0..0..0..2....0..2..1..0..2..0....0..2..2..2..1..1
%e A206411 ..2..0..2..2..1..0....0..0..0..2..2..2....1..0..2..1..0..0....0..2..0..2..2..1
%e A206411 ..1..2..1..2..2..1....2..2..0..2..0..2....2..1..0..2..1..1....1..0..0..0..2..2
%K A206411 nonn,new
%O A206411 1,1
%A A206411 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206410
%S A206410 15087,953865,62351943,4099250889,269845754715,17769571483479,
%T A206410 1170258128101905,77072446958972853,5075985987460513995,
%U A206410 334304949004000847991,22017374591980900445685,1450067958051192311462769
%N A206410 Number of (n+1)X5 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206410 Column 4 of A206414
%H A206410 R. H. Hardin, <a href="/A206410/b206410.txt">Table of n, a(n) for n = 1..210</a>
%F A206410 Empirical: a(n) = 121*a(n-1) -4842*a(n-2) +93442*a(n-3) -984794*a(n-4) +5503671*a(n-5) -9072726*a(n-6) -78684541*a(n-7) +535626405*a(n-8) -991587240*a(n-9) -2580309619*a(n-10) +15747001849*a(n-11) -20598090193*a(n-12) -41549984245*a(n-13) +164996602285*a(n-14) -124687904298*a(n-15) -276964848873*a(n-16) +640627801097*a(n-17) -268835295686*a(n-18) -615894146346*a(n-19) +919184816798*a(n-20) -316305834243*a(n-21) -338190799906*a(n-22) +415103347869*a(n-23) -166830909951*a(n-24) +1177218958*a(n-25) +23150659951*a(n-26) -7462678489*a(n-27) +119744345*a(n-28) +412765357*a(n-29) -80434289*a(n-30) -308634*a(n-31) +1758176*a(n-32) -205020*a(n-33) +8424*a(n-34) -108*a(n-35)
%e A206410 Some solutions for n=4
%e A206410 ..0..2..1..2..1....1..0..1..1..1....1..0..1..2..0....1..2..2..1..2
%e A206410 ..1..0..2..0..2....1..1..0..0..0....2..1..0..1..2....1..1..1..2..1
%e A206410 ..1..1..0..2..0....1..2..1..0..2....1..0..0..0..1....2..2..2..2..1
%e A206410 ..1..2..1..0..1....1..1..2..1..0....1..0..0..2..0....1..1..1..1..0
%e A206410 ..1..1..2..1..1....0..0..1..2..1....0..0..2..0..0....0..0..1..1..1
%K A206410 nonn,new
%O A206410 1,1
%A A206410 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206409
%S A206409 2457,69585,2073999,62351943,1876995285,56515649709,1701726564411,
%T A206409 51240524315871,1542900272261037,46458185448564093,
%U A206409 1398899935206593883,42122201449345396239,1268339366974860798957,38190899219543171363901
%N A206409 Number of (n+1)X4 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206409 Column 3 of A206414
%H A206409 R. H. Hardin, <a href="/A206409/b206409.txt">Table of n, a(n) for n = 1..210</a>
%F A206409 Empirical: a(n) = 47*a(n-1) -608*a(n-2) +3198*a(n-3) -5911*a(n-4) -7789*a(n-5) +44215*a(n-6) -38455*a(n-7) -50916*a(n-8) +86298*a(n-9) +5407*a(n-10) -54221*a(n-11) +10208*a(n-12) +11424*a(n-13) -2506*a(n-14) -502*a(n-15) +112*a(n-16)
%e A206409 Some solutions for n=4
%e A206409 ..1..1..1..1....0..1..2..0....2..1..1..1....0..0..1..2....1..2..0..2
%e A206409 ..1..0..1..2....1..0..1..2....2..1..1..2....1..0..1..1....1..1..2..2
%e A206409 ..1..0..0..1....1..1..2..0....1..1..1..1....0..2..0..0....1..2..2..1
%e A206409 ..2..1..0..0....0..1..2..2....1..2..2..1....0..0..1..0....2..0..2..2
%e A206409 ..0..2..1..0....2..0..1..1....2..2..1..0....2..2..0..1....2..0..0..2
%K A206409 nonn,new
%O A206409 1,1
%A A206409 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206408
%S A206408 423,5199,69585,953865,13148043,181467207,2505335985,34591087977,
%T A206408 477605748651,6594418668279,91050821905233,1257162198849945,
%U A206408 17357963791492635,239665900404460215,3309129149913889857
%N A206408 Number of (n+1)X3 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206408 Column 2 of A206414
%H A206408 R. H. Hardin, <a href="/A206408/b206408.txt">Table of n, a(n) for n = 1..210</a>
%F A206408 Empirical: a(n) = 19*a(n-1) -78*a(n-2) +86*a(n-3) +17*a(n-4) -41*a(n-5) +10*a(n-6)
%e A206408 Some solutions for n=4
%e A206408 ..2..2..0....0..0..1....2..2..2....2..0..1....0..0..1....2..0..0....1..2..0
%e A206408 ..1..2..2....1..1..0....0..0..2....0..0..0....0..2..0....0..0..1....2..1..2
%e A206408 ..2..1..2....2..1..0....1..0..0....2..2..0....1..0..0....1..0..0....2..1..2
%e A206408 ..1..0..1....2..2..1....1..0..0....0..2..0....1..0..0....1..0..2....1..2..2
%e A206408 ..1..1..1....2..1..1....2..1..0....0..0..2....2..1..1....2..1..0....1..2..0
%K A206408 nonn,new
%O A206408 1,1
%A A206408 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206407
%S A206407 81,423,2457,15087,94761,600519,3818649,24314127,154889673,986887623,
%T A206407 6288452889,40071132591,255342940521,1627113214023,10368413881497,
%U A206407 66070427765967,421019298884361,2682853284675399,17095895564336409
%N A206407 Number of (n+1)X2 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206407 Column 1 of A206414
%H A206407 R. H. Hardin, <a href="/A206407/b206407.txt">Table of n, a(n) for n = 1..210</a>
%F A206407 Empirical: a(n) = 9*a(n-1) -17*a(n-2) +a(n-3) +4*a(n-4)
%e A206407 Some solutions for n=4
%e A206407 ..1..2....1..2....1..2....0..1....2..0....1..2....1..2....2..1....2..2....1..2
%e A206407 ..0..0....1..1....0..1....2..0....2..2....1..2....0..2....0..2....0..0....0..1
%e A206407 ..1..2....1..1....2..0....2..2....2..0....1..2....0..1....1..0....2..2....2..0
%e A206407 ..0..2....1..0....2..0....0..2....2..2....1..1....0..2....2..1....0..2....2..2
%e A206407 ..0..1....1..0....0..1....2..0....0..2....2..1....1..2....1..0....2..0....0..0
%K A206407 nonn,new
%O A206407 1,1
%A A206407 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A204457
%S A204457 1,3,5,7,9,11,15,17,19,21,23,25,27,29,31,33,35,37,41,43,45,47,49,51,
%T A204457 53,55,57,59,61,63,67,69,71,73,75,77,79,81,83,85,87,89,93,95,97,99,
%U A204457 101,103,105,107,109
%N A204457 Odd numbers not divisible by 13.
%C A204457 For the general case of odd numbers not divisible by primes see a comment on A204454, where the o.g.f.s and the formulae in terms of floor functions are given.
%C A204457 The numerator polynomial of the o.g.f. given in the formula section has coefficients 1,2,2,2,2,2,4,2,2,2,2,2,1, see row no. 6 of A204456. The first seven numbers are the first differences of the sequence, starting with a(0)=0. The other numbers are obtained by mirroring around the center.
%H A204457 Reinhard Zumkeller, <a href="/A204457/b204457.txt">Table of n, a(n) for n = 1..10000</a>
%H A204457 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).
%F A204457 O.g.f.: x*(1 + 2*(x+x^6)*(1+x+x^2+x^3+x^4) + 4*x^6 + x^12)/((1-x^12)*(1-x)). The denominator can be factored.
%F A204457 a(n) = 2*n-1 + 2*floor((n+5)/12) = 2*n+1 + 2*floor((n-7)/12), n>=1. Note that this is -1 for n=0, but the o.g.f. starting with x^0 has a(0)=0.
%o A204457 (Haskell)
%o A204457 a204457 n = a204457_list !! (n-1)
%o A204457 a204457_list = [x | x <- [1, 3 ..], mod x 13 > 0]
%o A204457 -- Reinhard Zumkeller, Feb 08 2012
%o A204457 (PARI) a(n) = 2*n-1+(n+5)\12*2 \\ Charles R Greathouse IV, Feb 08 2012
%Y A204457 Cf. A204454 and cross-references there; A204458.
%K A204457 nonn,easy,new
%O A204457 1,2
%A A204457 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 07 2012

%I A206406
%S A206406 81,5199,2073999,4099250889,38883964087761,1772313933215113017,
%T A206406 388756746462491144596125,410796134340987248242931767083,
%U A206406 2092643007412378548348442168950591087
%N A206406 Number of (n+1)X(n+1) 0..2 arrays with every 2X3 or 3X2 subblock having an equal number of clockwise and anticlockwise edge increases
%C A206406 Diagonal of A206414
%e A206406 Some solutions for n=4
%e A206406 ..0..1..2..1..0....0..2..1..2..1....1..0..1..1..1....2..0..1..2..1
%e A206406 ..1..0..1..2..1....1..0..2..1..2....1..1..0..0..0....0..1..2..1..2
%e A206406 ..0..0..0..1..0....2..1..0..2..1....1..2..1..0..2....1..0..1..2..1
%e A206406 ..2..2..2..0..2....2..1..0..2..2....1..1..2..1..0....2..1..1..1..1
%e A206406 ..0..2..2..0..2....0..2..1..0..2....0..0..1..2..1....0..2..2..1..2
%K A206406 nonn,new
%O A206406 1,1
%A A206406 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A204458
%S A204458 1,3,5,7,9,11,13,15,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,
%T A204458 53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,87,89,91,93,95,97,99,
%U A204458 101,103,105,107,109,111,113,115,117,121,123,125,127,129,131,133,135,137,139,141
%N A204458 Odd numbers not divisible by 17.
%C A204458 For the general case of odd numbers not divisible by a prime see a comment on A204454. There the o.g.f.s and the formulae are given.
%C A204458 The numerator polynomial of the o.g.f. given below has coefficients 1,2,2,2,2,2,2,2,4,2,2,2,2,2,2,2,1. See the row no. 7 of the array A204456. The first nine numbers are the first differences of the sequence if one starts with a(0):=0. The remaining ones are obtained by mirroring around the central number 4.
%C A204458 Compare with A192861: certain numbers from here are missing there, like 35, 49, 53, 71, 89, 97, 99, .. and others are missing here like 51, 85, 119, ...
%F A204458 O.g.f.: x*(1 + x^16 + 2*x*(1+x^8)*sum(x^k,k=0..6) + 4*x^8)/((1-x^16)*(1-x)). The denominator can be factored.
%F A204458 a(n) = 2*n-1 + 2*floor((n+7)/16) = 2*n+1 + 2*floor((n-9)/16), n>=1. Note that for n=0 this is -1, but for the o.g.f. with start x^0 one uses a(0)=0.
%Y A204458 Cf. A204454, also for more crossrefs. A204457.
%K A204458 nonn,easy,new
%O A204458 1,2
%A A204458 Wolfdieter Lang (wolfdieter.lang(AT)kit.edu), Feb 07 2012

%I A206400
%S A206400 0,1,1,3,3,1,3,3,1,9,3,13,1,9,7,9,5,3,15,5,3,3,1,3,3,11,3,5,3,9,5,3,3,
%T A206400 19,1,3,13,5,5,3,9,5,3,3,5,9,3,15,5,7,11,13,9,33,1,9,3,5,13,9,5,3,3,
%U A206400 19,1,3,3,15,5,39,7,11,13,5,7,9,39,1,7,1,7
%N A206400 Number of composites of the form n^2 + 1 between two successive primes of this form.
%C A206400 a(n) = number of composites of A134406  between A002496(n) and A002496(n+1).
%H A206400 Michel Lagneau, <a href="/A206400/b206400.txt">Table of n, a(n) for n = 1..10000</a>
%e A206400 a(4) = 3 because there exists 3 composite numbers of the form n^2+1 : {50, 65, 82} between A002496(4) = 37 and A002496(5) = 101.
%p A206400 i:=0:for n from 2 to 1000 do:x:=n^2+1:if type (x,prime)=true then printf(`%d, `,i):i:=0:else i:=i+1:fi:od:
%o A206400 (PARI) c=0; for(n=2,1e9, !ispseudoprime(n^2+1) & c++ & next; print1(c","); c=0)  \\ M. F. Hasler, Feb 07 2012
%Y A206400 Cf. A002522, A002496, A134406.
%K A206400 nonn,new
%O A206400 1,4
%A A206400 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 07 2012

%I A206302
%S A206302 1,1,2,5,15,45,147,484,1649,5698,20055,71373,256998,933458,3418160,
%T A206302 12601291,46739440,174281272,652962532,2456802244,9279423954,
%U A206302 35170693226,133726116806,509926989456,1949634581725,7472405641631,28704432909043,110496066015970
%N A206302 G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{d|n} A(x^d)^(n/d).
%e A206302 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 45*x^5 + 147*x^6 +...
%e A206302 such that, by definition,
%e A206302 A(x) = 1 + x*A(x) + x^2*A(x)^2*A(x^2) + x^3*A(x)^3*A(x^3) + x^4*A(x)^4*A(x^2)^2*A(x^4) + x^5*A(x)^5*A(x^5) + x^6*A(x)^6*A(x^2)^3*A(x^3)^2*A(x^6) +...
%o A206302 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*exp(sumdiv(m,d, (m/d)*subst(log(A), x, x^d +x*O(x^n)))))); polcoeff(A, n)}
%o A206302 for(n=0, 35, print1(a(n), ", "))
%Y A206302 Cf. A206301.
%K A206302 nonn,new
%O A206302 0,3
%A A206302 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2012

%I A206334
%S A206334 3,5,7,10,12,15,16,18,19,23,25,26,27,28,29,30,33,34,36,38,39,40,41,42,
%T A206334 43,44,46,47,51,52,55,57,58,59,62,63,64,65,67,68,69,70,71,72,74,75,76,
%U A206334 77,80,83,84,85,86,87,88,89,90,91,93,95,96,97,103,104,105,106,107,109,115,119,122,123,124,125,126
%N A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.
%C A206334 n>3 is in the sequence just in case the elliptic curve y^2 = 4*x^4 + (n^2+8)*x^2 + 4 has positive rank.  Note that (0,2) is on that curve.
%C A206334 n is in the sequence just in case there are positive rational numbers x,y such that x*y>1 and x - 1/x + y - 1/y = n.
%C A206334 The triangle whose sides are [(4*k^6+8*k^5+8*k^4+4*k^3+2*k^2+2*k+1)/((k+1)*k*(2*k^2+2*k+1)), (4*k^6+16*k^5+28*k^4+28*k^3+18*k^2+6*k+1)/((k+1)*k*(2*k^2+2*k+1)), 4*k^2+4*k+4] has area equal to its third side.  Hence, starting with the second term, A112087 is a subsequence of the present sequence.
%C A206334 The triangle whose sides are [(k^6+2*k^4+k^2+1)/(k*(k^2+1)), (k^4+3*k^2+1)/(k*(k^2+1)), (k^2+2)*k] has area equal to its third side. Hence, starting with the first positive term, A054602 is a subsequence of the present sequence. [This subsequence found by Dragan K, see second link, below.]
%C A206334 The triangle whose sides are [(k^8+6*k^6+13*k^4+13*k^2+4)/(k*(k^2+2)*(k^2+1)), (k^6+3*k^4+5*k^2+4)/(k*(k^2+2)*(k^2+1)), k*(k^2+4)] has area equal to its third side. Hence A155965 is a subsequence of the present sequence.
%H A206334 James R. Buddenhagen, <a href="/A206334/a206334.txt">Table of triangles up to n=145</a>
%H A206334 Dragan K and Rita the dog, <a href="http://answers.yahoo.com/question/index;_ylt=As9iUvdShLradeXMqxZh.53sy6IX;_ylv=3?qid=20120127091629AACRNuT">Question and answer</a>
%H A206334 Ian Connell, <a href="http://www.math.mcgill.ca/connell/"> APECS elliptic curve software</a> (which runs under old versions of Maple).
%H A206334 Eric Weisstein, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">MathWorld: Heronian Triangle</a>.
%e A206334 5 is in the sequence because the triangle with sides (37/6, 13/6, 5) has area 5, one side 5, and the other two sides rational.
%Y A206334 Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).
%K A206334 nonn,new
%O A206334 1,1
%A A206334 James R. Buddenhagen (jbuddenh(AT)gmail.com), Feb 06 2012

%I A206305
%S A206305 1,2,56,4304,647552,161009408,59798825984,31018594543616,
%T A206305 21421107883900928,19000925396453752832,21053097631093130264576,
%U A206305 28496291901064818145624064
%N A206305 E.g.f. A(x)=sum(n>=0, a(n)*x^(2*n+1)/(2*n+1)! ) is inverse to f(x)=2*x-tan(x)
%F A206305 a(n)=(sum(k=1..2*n, (2*n+k)!*sum(j=1..k, (-1)^(j)/(k-j)!*((sum(l=0..j-1, (1/((j-l)!*l!)*sum(m=j-l..2*n-l+j, binomial(m-1,-l+j-1)*m!*2^(2*n-m+j)*(-1)^(n+m)*stirling2(2*n-l+j,m)))/(2*n-l+j)!)))))), n>0, a(0)=1.
%o A206305 (Maxima) a(n):=if n=0 then 1 else (sum((2*n+k)!*sum((-1)^(j)/(k-j)!*((sum((1/((j-l)!*l!)*sum(binomial(m-1,-l+j-1)*m!*2^(2*n-m+j)*(-1)^(n+m)*stirling2(2*n-l+j,m),m,j-l,2*n-l+j))/(2*n-l+j)!,l,0,j-1))),j,1,k),k,1,2*n));
%K A206305 nonn,new
%O A206305 0,2
%A A206305 Vladimir Kruchinin (kru(AT)ie.tusur.ru), Feb 06 2012

%I A206396
%S A206396 1,1,1,1,1,1,2,4,4,2,5,25,26,25,5,15,172,206,206,172,15,51,1201,1592,
%T A206396 1931,1592,1201,51,187,8404,12428,16784,16784,12428,8404,187,715,
%U A206396 58825,96632,151630,170796,151630,96632,58825,715,2795,411772,752552,1343560
%N A206396 T(n,k)=Number of nXk 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206396 Table starts
%C A206396 ...1.....1......1........2.........5.........15..........51..........187
%C A206396 ...1.....1......4.......25.......172.......1201........8404........58825
%C A206396 ...1.....4.....26......206......1592......12428.......96632.......752552
%C A206396 ...2....25....206.....1931.....16784.....151630.....1343560.....12046648
%C A206396 ...5...172...1592....16784....170796....1787258....18574298....193499878
%C A206396 ..15..1201..12428...151630...1787258...21983256...268956972...3301485294
%C A206396 ..51..8404..96632..1343560..18574298..268956972..3889732730..56960076094
%C A206396 .187.58825.752552.12046648.193499878.3301485294.56960076094.998388746378
%H A206396 R. H. Hardin, <a href="/A206396/b206396.txt">Table of n, a(n) for n = 1..127</a>
%e A206396 Some solutions for n=4 k=3
%e A206396 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206396 ..2..3..0....2..3..4....2..3..4....3..4..5....3..4..5....2..3..0....2..3..4
%e A206396 ..1..4..5....1..0..5....4..5..1....5..0..1....1..0..3....4..5..1....1..5..0
%e A206396 ..0..2..1....3..2..1....1..2..0....4..3..2....2..5..1....0..2..3....0..2..3
%Y A206396 Column 1 is A007581(n-3)
%Y A206396 Column 2 is A034494(n-2)
%K A206396 nonn,tabl,new
%O A206396 1,7
%A A206396 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206395
%S A206395 51,8404,96632,1343560,18574298,268956972,3889732730,56960076094,
%T A206395 835640037768,12318524591500
%N A206395 Number of nX7 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206395 Column 7 of A206396
%e A206395 Some solutions for n=4
%e A206395 ..0..1..2..0..1..2..3....0..1..2..3..4..0..1....0..1..2..3..0..1..2
%e A206395 ..2..3..4..5..3..4..5....2..3..5..0..1..2..4....2..3..4..1..5..3..4
%e A206395 ..4..5..1..2..0..1..2....5..4..1..2..5..3..0....4..5..0..2..4..0..5
%e A206395 ..0..2..3..4..5..3..4....0..2..3..4..0..1..5....0..1..3..5..1..2..3
%K A206395 nonn,new
%O A206395 1,1
%A A206395 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206394
%S A206394 15,1201,12428,151630,1787258,21983256,268956972,3301485294,
%T A206394 40603817198,499201392028,6147736324286,75698270712532,
%U A206394 932481394642188,11487787221488608,141538253304943642,1744036023952329892
%N A206394 Number of nX6 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206394 Column 6 of A206396
%H A206394 R. H. Hardin, <a href="/A206394/b206394.txt">Table of n, a(n) for n = 1..31</a>
%e A206394 Some solutions for n=4
%e A206394 ..0..1..2..0..1..3....0..1..2..0..3..4....0..1..2..0..3..1....0..1..2..0..1..2
%e A206394 ..2..3..4..5..2..4....2..5..4..1..2..5....2..3..4..1..2..4....2..3..4..5..3..4
%e A206394 ..4..5..0..1..3..5....1..3..0..5..4..0....1..5..0..3..5..0....5..0..1..2..0..1
%e A206394 ..0..1..2..4..0..1....5..4..2..3..1..2....0..4..1..2..4..1....3..4..5..3..4..2
%K A206394 nonn,new
%O A206394 1,1
%A A206394 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206393
%S A206393 5,172,1592,16784,170796,1787258,18574298,193499878,2014220262,
%T A206393 20973561786,218384365724,2274041419152,23679689583822,
%U A206393 246582133806618,2567714799158370,26738384205646006,278434494273903426
%N A206393 Number of nX5 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206393 Column 5 of A206396
%H A206393 R. H. Hardin, <a href="/A206393/b206393.txt">Table of n, a(n) for n = 1..210</a>
%F A206393 Empirical: a(n) = 19*a(n-1) -28*a(n-2) -1368*a(n-3) +6837*a(n-4) +30894*a(n-5) -259715*a(n-6) -158240*a(n-7) +4305294*a(n-8) -2907345*a(n-9) -36456187*a(n-10) +17700441*a(n-11) +264833772*a(n-12) +359253348*a(n-13) -3077852142*a(n-14) -4085375360*a(n-15) +29632825925*a(n-16) +13040334064*a(n-17) -175557464522*a(n-18) -60798023894*a(n-19) +1054573134761*a(n-20) +156059590354*a(n-21) -7365833614192*a(n-22) +9642172417687*a(n-23) +18919359102713*a(n-24) -74981349795094*a(n-25) +77760958965261*a(n-26) +59754494935940*a(n-27) -287857949600101*a(n-28) +691064089663409*a(n-29) -956324740445126*a(n-30) -738343162745785*a(n-31) +3909998344750953*a(n-32) -3355221285631341*a(n-33) -1701316400605256*a(n-34) +3179682162276913*a(n-35) -270478484532986*a(n-36) -5293044447668192*a(n-37) +25344266042370966*a(n-38) -28784566717360175*a(n-39) -17947836174493735*a(n-40) +67609956522748602*a(n-41) -101102908627491482*a(n-42) +104803692847485310*a(n-43) -57693328594689530*a(n-44) -10870560756068936*a(n-45) +69553561235094500*a(n-46) -80973326026485056*a(n-47) +47246917468719560*a(n-48) -11386765181702208*a(n-49) -2931176567684912*a(n-50) +4099580409184672*a(n-51) -2335307914381056*a(n-52) +605917226114688*a(n-53) +41498237271040*a(n-54) -93505834844160*a(n-55) +25819492058112*a(n-56) +690971830272*a(n-57) -51588218880*a(n-58) +2691366912*a(n-59) for n>63
%e A206393 Some solutions for n=4
%e A206393 ..0..1..2..3..0....0..1..2..3..4....0..1..2..0..1....0..1..2..3..4
%e A206393 ..2..3..4..5..1....4..5..0..1..5....2..3..4..5..2....3..4..0..5..1
%e A206393 ..5..0..1..2..3....1..2..3..4..0....1..5..0..3..4....5..2..1..4..0
%e A206393 ..3..2..5..4..0....3..4..1..5..2....0..4..1..2..5....4..0..3..2..5
%K A206393 nonn,new
%O A206393 1,1
%A A206393 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206392
%S A206392 2,25,206,1931,16784,151630,1343560,12046648,107401612,960610876,
%T A206392 8576723918,76644197120,684569775162,6116039277778,54633888067900,
%U A206392 488076198634430,4360092827131800,38950530867108738,347957262094712236
%N A206392 Number of nX4 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206392 Column 4 of A206396
%H A206392 R. H. Hardin, <a href="/A206392/b206392.txt">Table of n, a(n) for n = 1..210</a>
%F A206392 Empirical: a(n) = 14*a(n-1) -32*a(n-2) -281*a(n-3) +1738*a(n-4) -1728*a(n-5) -9767*a(n-6) +21963*a(n-7) -7410*a(n-8) +5706*a(n-9) -19704*a(n-10) +6028*a(n-11) +2512*a(n-12) -2176*a(n-13) for n>15
%e A206392 Some solutions for n=4
%e A206392 ..0..1..2..0....0..1..2..0....0..1..2..3....0..1..2..3....0..1..2..3
%e A206392 ..2..3..4..5....2..3..4..5....4..3..0..5....4..3..0..4....3..4..5..0
%e A206392 ..4..5..0..1....1..5..0..1....5..2..4..1....1..2..5..1....1..0..3..1
%e A206392 ..0..1..2..3....3..4..2..3....0..1..5..3....0..4..3..2....2..5..4..2
%K A206392 nonn,new
%O A206392 1,1
%A A206392 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206391
%S A206391 1,4,26,206,1592,12428,96632,752552,5856992,45595664,354918176,
%T A206391 2762807264,21506309504,167411006144,1303169775488,10144214859392,
%U A206391 78965192918528,614685592164608,4784872150020608,37246687385693696
%N A206391 Number of nX3 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206391 Column 3 of A206396
%H A206391 R. H. Hardin, <a href="/A206391/b206391.txt">Table of n, a(n) for n = 1..210</a>
%F A206391 Empirical: a(n) = 6*a(n-1) +18*a(n-2) -32*a(n-3) for n>5
%e A206391 Some solutions for n=4
%e A206391 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206391 ..3..4..5....3..4..5....2..3..4....2..3..0....2..3..4....3..4..5....2..3..0
%e A206391 ..5..0..1....1..0..3....4..5..1....4..5..1....1..5..0....1..0..3....1..4..5
%e A206391 ..4..3..2....2..5..1....3..2..0....0..2..4....4..2..3....4..5..1....5..0..1
%K A206391 nonn,new
%O A206391 1,2
%A A206391 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206390
%S A206390 1,1,26,1931,170796,21983256,3889732730,998388746378
%N A206390 Number of nXn 0..5 arrays with no element equal to another within a city block distance of two, and new values 0..5 introduced in row major order
%C A206390 Diagonal of A206396
%e A206390 Some solutions for n=4
%e A206390 ..0..1..2..0....0..1..2..3....0..1..2..0....0..1..2..3....0..1..2..3
%e A206390 ..2..3..4..1....3..4..5..0....2..3..4..5....2..3..4..5....4..3..5..4
%e A206390 ..1..5..0..2....1..0..3..1....4..5..0..1....4..5..0..1....5..2..0..1
%e A206390 ..4..2..3..4....2..5..4..2....0..1..2..3....3..1..2..3....0..1..4..3
%K A206390 nonn,new
%O A206390 1,3
%A A206390 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206389
%S A206389 1,1,1,1,1,1,2,4,4,2,5,33,55,33,5,15,380,1368,1368,380,15,52,4801,
%T A206389 34442,67716,34442,4801,52,202,62004,868994,3328979,3328979,868994,
%U A206389 62004,202,855,804833,21916090,164270604,319902496,164270604,21916090,804833,855
%N A206389 T(n,k)=Number of nXk 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206389 Table starts
%C A206389 ...1......1.........1..........2...........5..........15............52
%C A206389 ...1......1.........4.........33.........380........4801.........62004
%C A206389 ...1......4........55.......1368.......34442......868994......21916090
%C A206389 ...2.....33......1368......67716.....3328979...164270604....8094257014
%C A206389 ...5....380.....34442....3328979...319902496.30809397399.2965468376203
%C A206389 ..15...4801....868994..164270604.30809397399
%C A206389 ..52..62004..21916090.8094257014
%C A206389 .202.804833.552801398
%H A206389 R. H. Hardin, <a href="/A206389/b206389.txt">Table of n, a(n) for n = 1..60</a>
%e A206389 Some solutions for n=4 k=3
%e A206389 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206389 ..2..3..0....2..3..0....3..4..0....2..3..4....2..3..4....3..4..5....2..3..4
%e A206389 ..4..5..1....4..5..6....1..2..5....4..5..1....1..5..6....5..2..1....4..5..6
%e A206389 ..6..2..4....6..2..1....6..0..1....6..0..2....6..2..0....1..6..4....6..0..1
%Y A206389 Column 1 is A056272(n-2)
%Y A206389 Column 2 is A198900(n-1)
%K A206389 nonn,tabl,new
%O A206389 1,7
%A A206389 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206388
%S A206388 52,62004,21916090,8094257014,2965468376203
%N A206388 Number of nX7 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206388 Column 7 of A206389
%e A206388 Some solutions for n=4
%e A206388 ..0..1..2..0..1..2..3....0..1..2..3..0..2..1....0..1..2..0..1..2..0
%e A206388 ..4..3..5..6..3..4..5....4..3..5..4..6..5..0....3..4..5..3..6..4..1
%e A206388 ..1..6..4..1..0..6..1....1..2..6..0..1..3..6....1..2..6..4..2..3..5
%e A206388 ..2..0..3..5..2..3..4....3..4..1..5..2..0..4....4..3..0..5..1..6..4
%K A206388 nonn,new
%O A206388 1,1
%A A206388 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206387
%S A206387 15,4801,868994,164270604,30809397399
%N A206387 Number of nX6 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206387 Column 6 of A206389
%e A206387 Some solutions for n=4
%e A206387 ..0..1..2..0..3..1....0..1..2..0..1..2....0..1..2..3..4..1....0..1..2..3..0..1
%e A206387 ..4..3..5..6..2..0....2..3..4..5..6..3....2..3..4..0..5..6....2..3..0..1..2..3
%e A206387 ..2..6..4..1..5..3....6..5..0..3..2..4....5..6..1..2..3..0....1..4..5..6..4..5
%e A206387 ..3..5..0..2..4..6....3..1..6..4..5..6....3..0..5..6..1..2....3..6..1..0..3..6
%K A206387 nonn,new
%O A206387 1,1
%A A206387 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206386
%S A206386 5,380,34442,3328979,319902496,30809397399,2965468376203,
%T A206386 285477006510014,27481270649615774
%N A206386 Number of nX5 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206386 Column 5 of A206389
%e A206386 Some solutions for n=4
%e A206386 ..0..1..2..3..0....0..1..2..0..1....0..1..2..0..1....0..1..2..0..1
%e A206386 ..2..3..0..1..2....2..3..4..5..2....2..3..4..5..6....2..3..4..5..6
%e A206386 ..1..4..5..6..3....1..0..6..3..4....6..0..1..2..0....4..5..0..1..2
%e A206386 ..6..0..1..4..5....3..2..5..0..6....1..2..3..6..1....0..1..6..3..4
%K A206386 nonn,new
%O A206386 1,1
%A A206386 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206385
%S A206385 2,33,1368,67716,3328979,164270604,8094257014,399073153303,
%T A206385 19671060879306,969710249315342,47801400299066037,2356379708874988482,
%U A206385 116157573684573753928,5725991875022057193833,282262740672653624738188
%N A206385 Number of nX4 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206385 Column 4 of A206389
%H A206385 R. H. Hardin, <a href="/A206385/b206385.txt">Table of n, a(n) for n = 1..29</a>
%e A206385 Some solutions for n=4
%e A206385 ..0..1..2..0....0..1..2..3....0..1..2..3....0..1..2..3....0..1..2..0
%e A206385 ..2..3..4..5....3..4..5..6....2..3..4..5....2..3..4..5....2..3..4..5
%e A206385 ..1..5..6..3....6..2..0..1....1..5..6..0....4..5..0..6....5..6..0..1
%e A206385 ..0..2..1..4....0..1..3..2....3..4..1..2....6..1..3..4....4..1..5..6
%K A206385 nonn,new
%O A206385 1,1
%A A206385 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206384
%S A206384 1,4,55,1368,34442,868994,21916090,552801398,13943026782,351682327314,
%T A206384 8870382111250,223735362633678,5643217340398022,142337380859578634,
%U A206384 3590138049351706810,90553101869397071558,2283996915742293943662
%N A206384 Number of nX3 0..6 arrays with no element equal to another within a city block distance of two, and new values 0..6 introduced in row major order
%C A206384 Column 3 of A206389
%H A206384 R. H. Hardin, <a href="/A206384/b206384.txt">Table of n, a(n) for n = 1..150</a>
%F A206384 Empirical: a(n) = 21*a(n-1) +131*a(n-2) -637*a(n-3) +486*a(n-4) for n>6
%e A206384 Some solutions for n=4
%e A206384 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206384 ..3..4..0....2..3..4....2..3..4....2..3..4....2..3..4....2..3..0....3..4..5
%e A206384 ..1..2..5....4..5..6....4..5..6....4..5..0....1..0..5....1..4..5....5..2..1
%e A206384 ..6..0..1....3..0..1....3..1..0....3..1..6....4..2..1....6..0..1....1..6..4
%K A206384 nonn,new
%O A206384 1,2
%A A206384 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206383
%S A206383 15,85,85,641,1369,641,5257,25945,25945,5257,44585,520097,1198601,
%T A206383 520097,44585,385465,10690969,57656713,57656713,10690969,385465,
%U A206383 3375401,222329489,2808967145,6554439373,2808967145,222329489,3375401,29817817
%N A206383 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206383 Table starts
%C A206383 .......15..........85.............641................5257
%C A206383 .......85........1369...........25945..............520097
%C A206383 ......641.......25945.........1198601............57656713
%C A206383 .....5257......520097........57656713..........6554439373
%C A206383 ....44585....10690969......2808967145........748963157161
%C A206383 ...385465...222329489....137393517193......85678539791333
%C A206383 ..3375401..4648291369...6728546532521....9803619036105721
%C A206383 .29817817.97418918945.329641979105737.1121819841273065957
%H A206383 R. H. Hardin, <a href="/A206383/b206383.txt">Table of n, a(n) for n = 1..84</a>
%e A206383 Some solutions for n=4 k=3
%e A206383 ..0..0..0..0....0..0..0..0....0..0..1..2....0..0..0..1....0..1..0..2
%e A206383 ..1..0..1..0....0..0..0..0....0..2..0..1....0..1..0..0....2..1..3..2
%e A206383 ..1..1..1..1....0..0..0..0....2..1..2..0....0..0..3..0....0..1..3..1
%e A206383 ..1..0..1..2....0..0..0..0....0..2..2..2....3..0..0..0....3..1..0..0
%e A206383 ..1..1..1..1....0..0..0..0....1..0..2..0....2..3..0..1....3..1..2..3
%Y A206383 Column 1 is A206170
%K A206383 nonn,new
%O A206383 1,1
%A A206383 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206382
%S A206382 3375401,4648291369,6728546532521,9803619036105721,
%T A206382 14294968502144130953,20848171803769792984489
%N A206382 Number of (n+1)X8 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206382 Column 7 of A206383
%e A206382 Some solutions for n=4
%e A206382 ..0..1..1..2..2..0..1..3....0..0..0..0..0..1..2..1....0..0..0..0..1..0..1..0
%e A206382 ..1..3..1..1..2..2..0..1....1..1..2..2..3..1..2..3....0..1..0..1..1..1..1..1
%e A206382 ..1..1..2..1..1..2..2..0....3..3..0..0..3..0..2..3....1..1..1..3..1..2..1..2
%e A206382 ..2..1..1..1..2..2..1..2....1..2..1..1..3..1..1..0....3..1..0..1..0..1..3..1
%e A206382 ..1..3..1..3..1..2..2..2....3..2..3..0..2..2..2..0....2..3..1..2..1..3..3..3
%K A206382 nonn,new
%O A206382 1,1
%A A206382 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206381
%S A206381 385465,222329489,137393517193,85678539791333,53482634218458697,
%T A206381 33391295169652273181,20848171803769792984489,
%U A206381 13016855431061639609291213,8127274925445989337637081801,5074392067217049794271345475469
%N A206381 Number of (n+1)X7 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206381 Column 6 of A206383
%H A206381 R. H. Hardin, <a href="/A206381/b206381.txt">Table of n, a(n) for n = 1..26</a>
%e A206381 Some solutions for n=4
%e A206381 ..0..0..0..0..0..0..0....0..0..1..0..0..0..2....0..0..1..1..2..2..3
%e A206381 ..0..1..0..1..0..2..0....3..0..0..0..2..0..0....2..2..2..3..3..0..3
%e A206381 ..3..0..3..0..2..1..2....2..3..0..1..0..3..0....1..1..0..0..2..0..2
%e A206381 ..1..3..2..3..0..2..0....0..2..3..0..3..3..3....2..3..2..1..1..3..2
%e A206381 ..2..1..3..1..3..0..1....3..0..2..3..2..3..2....2..3..0..3..0..3..2
%K A206381 nonn,new
%O A206381 1,1
%A A206381 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206380
%S A206380 44585,10690969,2808967145,748963157161,200108001096137,
%T A206380 53482634218458697,14294968502144130953,3820826066766517546921,
%U A206380 1021249666781298003730505,272964826866462882832853641
%N A206380 Number of (n+1)X6 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206380 Column 5 of A206383
%H A206380 R. H. Hardin, <a href="/A206380/b206380.txt">Table of n, a(n) for n = 1..72</a>
%F A206380 Empirical: a(n) = 394*a(n-1) -38562*a(n-2) +1246392*a(n-3) +6334675*a(n-4) -1215511426*a(n-5) +25096715040*a(n-6) -111433495776*a(n-7) -2633554263696*a(n-8) +38447016473440*a(n-9) -124529929487104*a(n-10) -973109346072576*a(n-11) +8528195948079360*a(n-12) -13461810857735680*a(n-13) -67310021223342080*a(n-14) +238037998037753856*a(n-15) +44544280591749120*a(n-16) -947327372288188416*a(n-17) +643157082054918144*a(n-18) +987246769084563456*a(n-19) -892353207989698560*a(n-20)
%e A206380 Some solutions for n=4
%e A206380 ..0..0..0..0..1..2....0..0..0..0..0..0....0..0..0..0..1..1....0..1..2..1..1..0
%e A206380 ..3..3..1..3..3..0....0..0..0..0..0..0....0..2..0..1..1..0....0..1..0..3..2..2
%e A206380 ..1..2..0..2..1..2....0..0..0..0..0..0....2..2..2..0..1..1....2..1..0..3..0..1
%e A206380 ..1..2..1..2..1..3....0..0..0..0..0..0....0..2..0..0..0..1....0..1..0..3..2..2
%e A206380 ..3..2..3..0..0..2....0..0..0..0..0..0....2..1..2..0..1..1....2..1..2..3..0..1
%K A206380 nonn,new
%O A206380 1,1
%A A206380 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206379
%S A206379 5257,520097,57656713,6554439373,748963157161,85678539791333,
%T A206379 9803619036105721,1121819841273065957,128370307988576256937,
%U A206379 14689502396007272718677,1680930696757228231925017
%N A206379 Number of (n+1)X5 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206379 Column 4 of A206383
%H A206379 R. H. Hardin, <a href="/A206379/b206379.txt">Table of n, a(n) for n = 1..107</a>
%F A206379 Empirical: a(n) = 160*a(n-1) -5537*a(n-2) +25556*a(n-3) +1495902*a(n-4) -23370552*a(n-5) +81849864*a(n-6) +544237824*a(n-7) -5119960320*a(n-8) +15199301376*a(n-9) -19180578816*a(n-10) +8497004544*a(n-11)
%e A206379 Some solutions for n=4
%e A206379 ..0..1..1..1..0....0..1..0..0..0....0..1..0..0..0....0..0..1..0..2
%e A206379 ..2..2..3..2..3....1..1..1..0..2....1..1..1..0..3....1..2..1..3..2
%e A206379 ..3..0..0..0..1....3..1..2..1..0....2..1..2..1..0....0..3..0..0..2
%e A206379 ..3..1..1..2..1....1..2..2..2..1....2..2..2..2..1....1..1..1..1..1
%e A206379 ..3..0..3..0..3....0..1..2..0..2....0..2..0..2..2....2..2..2..2..0
%K A206379 nonn,new
%O A206379 1,1
%A A206379 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206378
%S A206378 641,25945,1198601,57656713,2808967145,137393517193,6728546532521,
%T A206378 329641979105737,16151593118685929,791414923478310409,
%U A206378 38779131391953424169,1900174398169760171593,93108499268211138032489
%N A206378 Number of (n+1)X4 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206378 Column 3 of A206383
%H A206378 R. H. Hardin, <a href="/A206378/b206378.txt">Table of n, a(n) for n = 1..210</a>
%F A206378 Empirical: a(n) = 74*a(n-1) -1401*a(n-2) +9064*a(n-3) -21992*a(n-4) +21312*a(n-5) -7056*a(n-6)
%e A206378 Some solutions for n=4
%e A206378 ..0..0..0..1....0..0..0..0....0..0..1..1....0..1..0..2....0..0..0..1
%e A206378 ..2..3..3..1....0..1..0..1....0..2..0..1....2..1..3..2....0..1..0..0
%e A206378 ..0..0..2..0....1..2..1..2....1..0..3..0....0..1..3..1....2..0..0..3
%e A206378 ..2..1..1..3....3..1..1..1....3..1..0..0....3..1..0..0....1..2..0..0
%e A206378 ..2..3..2..2....0..3..1..3....2..3..1..0....3..1..2..3....0..1..2..0
%K A206378 nonn,new
%O A206378 1,1
%A A206378 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206377
%S A206377 85,1369,25945,520097,10690969,222329489,4648291369,97418918945,
%T A206377 2043949959289,42905465857457,900849319640905,18916269750894785,
%U A206377 397226844350640025,8341623469422864209,175172765404628220265
%N A206377 Number of (n+1)X3 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206377 Column 2 of A206383
%H A206377 R. H. Hardin, <a href="/A206377/b206377.txt">Table of n, a(n) for n = 1..210</a>
%F A206377 Empirical: a(n) = 36*a(n-1) -377*a(n-2) +1398*a(n-3) -2064*a(n-4) +1008*a(n-5)
%e A206377 Some solutions for n=4
%e A206377 ..0..0..1....0..0..0....0..0..0....0..0..1....0..0..0....0..0..0....0..0..0
%e A206377 ..0..1..1....0..1..0....1..1..1....1..2..2....1..0..1....1..0..1....0..1..0
%e A206377 ..0..0..1....3..0..0....0..0..0....1..3..0....1..1..1....0..2..0....1..1..1
%e A206377 ..0..2..0....0..1..0....1..1..1....1..2..0....0..1..3....3..0..3....0..1..2
%e A206377 ..1..0..0....2..0..2....2..2..2....0..2..3....1..3..2....2..3..3....1..3..1
%K A206377 nonn,new
%O A206377 1,1
%A A206377 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206376
%S A206376 15,1369,1198601,6554439373,200108001096137,33391295169652273181
%N A206376 Number of (n+1)X(n+1) 0..3 arrays with every 2X2 subblock having the same number of equal diagonal or antidiagonal elements, and new values 0..3 introduced in row major order
%C A206376 Diagonal of A206383
%e A206376 Some solutions for n=4
%e A206376 ..0..0..0..1..0....0..0..0..0..1....0..1..0..1..0....0..0..0..0..0
%e A206376 ..0..2..0..0..2....1..2..2..2..3....0..0..2..0..0....1..0..1..0..2
%e A206376 ..3..0..0..1..0....1..3..1..0..3....3..0..0..1..0....3..1..1..1..0
%e A206376 ..3..3..0..0..2....0..2..1..0..2....1..3..0..0..0....1..0..1..0..0
%e A206376 ..1..3..3..0..0....3..3..3..3..1....0..1..3..0..2....3..1..1..1..0
%K A206376 nonn,new
%O A206376 1,1
%A A206376 R. H. Hardin (rhhardin(AT)att.net) Feb 07 2012

%I A206370
%S A206370 1,1,2,3,6,8,14,20,76,87,128,210,810,923,2646
%N A206370 Relatd to number of cyclotomic classes in GF(2^(2n)).
%D A206370 Jean-Pierre Flori, Fonctions booleennes, courbes algebriques et multiplication complexe, Thesis, ParisTech, Feb 03 2012; http://www.infres.enst.fr/~flori/thesis/thesis.pdf. See table on page 230.
%K A206370 nonn,more,new
%O A206370 2,3
%A A206370 N. J. A. Sloane (njas(AT)research.att.com), Feb 06 2012

%I A206368
%S A206368 1,20,116,135,171,194,740,1220,1419,1803,1892,1952,2696,3705,4575,
%T A206368 5186,7868,10659,11247,17948,18507,18548,19107,25545,27405,29294,
%U A206368 33500,34371,37820,48872,49184,53108,54620,58652,61760,67220,102296,104139,105908,113576
%N A206368 Numbers n such that A206369(n) = A206039(n+1).
%C A206368 A206475(a(n)) = 0. [Reinhard Zumkeller, Feb 08 2012]
%D A206368 L. Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842, 2011
%H A206368 Reinhard Zumkeller, <a href="/A206368/b206368.txt">Table of n, a(n) for n = 1..100</a>
%H A206368 László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>
%o A206368 (Haskell)
%o A206368 import Data.List (elemIndices)
%o A206368 a206368 n = a206368_list !! (n-1)
%o A206368 a206368_list = map (+ 1) $ elemIndices 0 a206475_list
%o A206368 -- Reinhard Zumkeller, Feb 08 2012
%K A206368 nonn,new
%O A206368 1,2
%A A206368 N. J. A. Sloane (njas(AT)research.att.com), Feb 06 2012

%I A206369
%S A206369 1,1,2,3,4,2,6,5,7,4,10,6,12,6,8,11,16,7,18,12,12,10,22,10,21,12,20,
%T A206369 18,28,8,30,21,20,16,24,21,36,18,24,20,40,12,42,30,28,22,46,22,43,21,
%U A206369 32,36,52,20,40,30,36,28,58,24,60,30,42,43,48,20,66,48,44,24,70,35
%N A206369 a(p^k)=p^k-p^(k-1)+p^(k-2)-...+-1. then extend by multiplicativity.
%C A206369 For more information see the Comments in A061020.
%D A206369 L. Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842, 2011. This is the function beta(n).
%H A206369 Reinhard Zumkeller, <a href="/A206369/b206369.txt">Table of n, a(n) for n = 1..10000</a>
%H A206369 László Tóth, <a href="http://arxiv.org/abs/1111.4842">A survey of the alternating sum-of-divisors function</a>
%o A206369 (Haskell)
%o A206369 a206369 n = product $
%o A206369    zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
%o A206369            h p e = sum $ take (fromInteger e + 1) $
%o A206369                          iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
%o A206369 -- Reinhard Zumkeller, Feb 08 2012
%Y A206369 a(n) = |A061020(n)|. Cf. A206368.
%Y A206369 Cf. A027748_row, A124010, A206475 (first differences).
%K A206369 nonn,mult,new
%O A206369 1,3
%A A206369 N. J. A. Sloane (njas(AT)research.att.com), Feb 06 2012

%I A206367
%S A206367 3,15,18,36,72,255
%N A206367 (2,3)-imperfect numbers.
%D A206367 L. Toth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842, 2011
%K A206367 nonn,new
%O A206367 1,1
%A A206367 N. J. A. Sloane (njas(AT)research.att.com), Feb 06 2012

%I A206366
%S A206366 15,151,151,1990,6329,1990,27548,294983,294983,27548,383021,13804578,
%T A206366 47155911,13804578,383021,5327971,643832706,7524874363,7524874363,
%U A206366 643832706,5327971,74147694,30049504156,1197873588223,4089691428777
%N A206366 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206366 Table starts
%C A206366 .........15............151..............1990...............27548
%C A206366 ........151...........6329............294983............13804578
%C A206366 .......1990.........294983..........47155911..........7524874363
%C A206366 ......27548.......13804578........7524874363.......4089691428777
%C A206366 .....383021......643832706.....1197873588223....2216692870846655
%C A206366 ....5327971....30049504156...190817902385518.1202272856786461731
%C A206366 ...74147694..1402595823865.30394990249581088
%C A206366 .1031960617.65463971182963
%H A206366 R. H. Hardin, <a href="/A206366/b206366.txt">Table of n, a(n) for n = 1..49</a>
%e A206366 Some solutions for n=4 k=3
%e A206366 ..0..0..1..1....0..0..0..0....0..0..0..0....0..0..0..0....0..1..1..0
%e A206366 ..2..1..2..0....1..2..3..2....1..2..1..2....1..2..1..2....0..1..2..1
%e A206366 ..2..0..3..0....2..1..2..1....0..0..0..0....0..1..2..0....3..2..1..1
%e A206366 ..0..2..1..1....1..3..2..0....3..1..2..3....3..3..0..1....1..0..3..2
%e A206366 ..0..2..1..0....3..3..0..1....1..0..3..2....0..1..2..1....3..3..0..0
%K A206366 nonn,tabl,new
%O A206366 1,1
%A A206366 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206365
%S A206365 383021,643832706,1197873588223,2216692870846655,4089300179444311432
%N A206365 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206365 Column 5 of A206366
%e A206365 Some solutions for n=4
%e A206365 ..0..0..0..0..1..2....0..0..0..0..0..0....0..0..0..0..1..0....0..0..0..0..0..0
%e A206365 ..1..2..3..1..3..3....1..2..3..2..3..1....2..1..2..3..3..3....1..2..3..2..1..2
%e A206365 ..0..1..3..2..1..1....0..1..0..3..0..3....3..3..2..1..0..1....0..3..0..0..1..3
%e A206365 ..2..1..2..0..2..0....3..1..3..1..0..0....3..0..1..2..0..2....1..3..0..2..0..0
%e A206365 ..3..0..0..2..1..1....3..0..1..1..2..1....0..1..0..2..1..1....2..1..2..3..1..0
%K A206365 nonn,new
%O A206365 1,1
%A A206365 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206364
%S A206364 27548,13804578,7524874363,4089691428777,2216692870846655,
%T A206364 1202272856786461731,652050586490715445332,353635210395974181372271,
%U A206364 191792723232477769611110455,104017873871839347470212875467
%N A206364 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206364 Column 4 of A206366
%H A206364 R. H. Hardin, <a href="/A206364/b206364.txt">Table of n, a(n) for n = 1..14</a>
%e A206364 Some solutions for n=4
%e A206364 ..0..0..0..1..0....0..0..1..0..0....0..0..0..0..1....0..0..0..1..0
%e A206364 ..1..2..1..2..0....0..1..0..1..1....1..2..3..1..2....1..2..1..3..1
%e A206364 ..0..1..0..3..2....2..0..2..2..3....3..2..0..1..1....3..0..0..1..0
%e A206364 ..1..1..0..0..3....0..3..2..3..0....2..3..0..2..3....2..0..1..0..2
%e A206364 ..1..3..1..1..0....2..3..0..1..1....1..3..1..0..3....0..1..2..1..1
%K A206364 nonn,new
%O A206364 1,1
%A A206364 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206363
%S A206363 1990,294983,47155911,7524874363,1197873588223,190817902385518,
%T A206363 30394990249581088,4841446696348626725,771174575648253890057,
%U A206363 122837158924767156108471,19566207413690834344037159
%N A206363 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206363 Column 3 of A206366
%H A206363 R. H. Hardin, <a href="/A206363/b206363.txt">Table of n, a(n) for n = 1..210</a>
%F A206363 Empirical: a(n) = 156*a(n-1) -1413*a(n-2) +329106*a(n-3) -4154064*a(n-4) +186840087*a(n-5) -8192746333*a(n-6) +38256252573*a(n-7) +1524493608637*a(n-8) -14650534294939*a(n-9) -70737673568729*a(n-10) +1266959328258358*a(n-11) +1305358080627014*a(n-12) -57723840510391912*a(n-13) -143071119831815622*a(n-14) +3140645711858022401*a(n-15) -1305066715344231552*a(n-16) -65229690816553397761*a(n-17) +75497968365687199186*a(n-18) +631869450113827724563*a(n-19) -1008205063597125759852*a(n-20) -17929402222374322998733*a(n-21) +122327054387103519125111*a(n-22) +464725801000735333183598*a(n-23) -2465472225898448207231288*a(n-24) -156547751115906706500371*a(n-25) -30698647352213321127103802*a(n-26) -169321220443896083823109141*a(n-27) +1024989169658009258328131623*a(n-28) -446176112972097179334633604*a(n-29) +3438194787300589043171190374*a(n-30) +53754059779526411972793401792*a(n-31) -99297157343228786341710316519*a(n-32) -186348746017147456654042037043*a(n-33) -980511766094394050160296684141*a(n-34) -3860545763955432766686121360048*a(n-35) +1800658275478467059082507907630*a(n-36) +5683480556604232205159834488193*a(n-37) +84436913072511515434531261294259*a(n-38) +201013788648537453968005814761691*a(n-39) +317143053951493206349513023222952*a(n-40) +465414406361574321457329730858706*a(n-41) -1774838715783548998747399345859554*a(n-42) -5386485181143264155006607744277576*a(n-43) -18430124632356464460977373517297142*a(n-44) -23176378783105043807857486557822660*a(n-45) -20265402938979906164077466135001889*a(n-46) +72214616284983713771584870800105104*a(n-47) +206109451346604589085910677991731511*a(n-48) +295665513674613843340094829344781018*a(n-49) +300995727057338655304477996783095257*a(n-50) -474763553278461043092075823566948470*a(n-51) +954925001289007754801210834954608087*a(n-52) -265048063242935544705538978751136419*a(n-53) +4033613235448021689216936549526879844*a(n-54) +2328665429101220413795440019452500581*a(n-55) +10817496708615259100742210303409406988*a(n-56) +6523544352168070916759371713077465450*a(n-57) -10911703961725738105473020634675713780*a(n-58) -52193695973520841774898350977964973300*a(n-59) -35808642699687643575940299358141412296*a(n-60) -27383986136907950207446881762894898920*a(n-61) -46431458138215326166908082652669157056*a(n-62) -30352758852440647468320897326795394656*a(n-63) +57336456437583528121359131409473443904*a(n-64) +256113389527547637754347868711690622592*a(n-65) +16050973371501943567388377553943270400*a(n-66) -462290927134426426834247835984190160896*a(n-67) +124450859935497506530885347185361995776*a(n-68) +545570025581283201732334566502703677440*a(n-69) -494641588038546120478581739110369755136*a(n-70) -262694706278462839542389619331891789824*a(n-71) +534467659856189181706516380893280731136*a(n-72) -114871871918666790476160929116685598720*a(n-73) -204484152643211333062197003990866067456*a(n-74) +216450190885624277290287175845323735040*a(n-75) -22914412904956493121107620055312498688*a(n-76) -100566381423434781578461610471596228608*a(n-77) +43503459079164257382178958052251664384*a(n-78) +7292513991812167577018793791078793216*a(n-79) -15644270164788059197341510897915396096*a(n-80) +8052695477492562845196427289715277824*a(n-81) +3143716862083141634766200203043143680*a(n-82) -2495101350879765737743335206525337600*a(n-83) -320589669443445488711333169856512000*a(n-84) +234029489504704351561662081269760000*a(n-85)
%e A206363 Some solutions for n=4
%e A206363 ..0..0..0..0....0..1..1..0....0..0..0..0....0..0..0..1....0..0..1..1
%e A206363 ..1..2..1..2....1..1..2..3....1..2..1..2....1..0..1..0....2..1..2..0
%e A206363 ..0..1..2..0....1..0..1..3....0..0..0..0....2..1..2..0....2..0..3..0
%e A206363 ..3..3..0..1....0..1..1..2....3..1..2..3....1..3..2..3....0..2..1..1
%e A206363 ..0..1..2..1....1..3..3..0....1..0..3..2....1..3..0..2....0..2..1..0
%K A206363 nonn,new
%O A206363 1,1
%A A206363 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206362
%S A206362 151,6329,294983,13804578,643832706,30049504156,1402595823865,
%T A206362 65463971182963,3055453674984819,142609963970592832,
%U A206362 6656158355663856332,310668667499188649660,14500109457714755675679,676776236342576687653295
%N A206362 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206362 Column 2 of A206366
%H A206362 R. H. Hardin, <a href="/A206362/b206362.txt">Table of n, a(n) for n = 1..210</a>
%F A206362 Empirical: a(n) = 45*a(n-1) +16*a(n-2) +3895*a(n-3) -48177*a(n-4) +77462*a(n-5) +110913*a(n-6) +1127659*a(n-7) -4335582*a(n-8) +2941967*a(n-9) -8832158*a(n-10) +22085991*a(n-11) -20203605*a(n-12) +43969692*a(n-13) -43415473*a(n-14) +53063555*a(n-15) -64756263*a(n-16) +19624030*a(n-17) -46916214*a(n-18) +4063848*a(n-19) +972000*a(n-20) +9782208*a(n-21)
%e A206362 Some solutions for n=4
%e A206362 ..0..0..0....0..0..1....0..1..1....0..0..0....0..0..0....0..1..1....0..1..0
%e A206362 ..1..2..1....2..1..2....0..2..0....1..2..1....1..0..1....2..1..2....0..0..1
%e A206362 ..0..1..3....0..0..1....3..3..0....1..3..1....0..1..0....3..0..1....0..1..2
%e A206362 ..1..3..3....0..3..2....1..3..2....0..0..1....1..2..0....2..0..0....1..2..0
%e A206362 ..1..2..0....1..2..0....2..0..1....0..2..3....3..3..0....1..3..0....2..2..1
%K A206362 nonn,new
%O A206362 1,1
%A A206362 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206361
%S A206361 15,151,1990,27548,383021,5327971,74147694,1031960617,14362233044,
%T A206361 199885520104,2781900205987,38716999664337,538842441505438,
%U A206361 7499320389877491,104371523732504906,1452586950516895892
%N A206361 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206361 Column 1 of A206366
%H A206361 R. H. Hardin, <a href="/A206361/b206361.txt">Table of n, a(n) for n = 1..210</a>
%F A206361 Empirical: a(n) = 16*a(n-1) -42*a(n-2) +234*a(n-3) -784*a(n-4) +714*a(n-5) -609*a(n-6) +1008*a(n-7) for n>8
%e A206361 Some solutions for n=4
%e A206361 ..0..0....0..0....0..0....0..0....0..0....0..0....0..0....0..0....0..0....0..0
%e A206361 ..0..1....1..1....0..1....0..1....1..0....1..1....0..1....1..1....1..1....0..1
%e A206361 ..1..0....0..0....1..2....2..3....0..1....0..0....1..0....2..0....1..0....1..0
%e A206361 ..1..0....1..0....0..2....0..1....1..0....0..1....1..1....1..0....0..2....1..0
%e A206361 ..2..1....0..1....2..1....3..1....2..2....1..2....2..2....0..1....3..3....0..1
%K A206361 nonn,new
%O A206361 1,1
%A A206361 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206360
%S A206360 15,6329,47155911,4089691428777,4089300179444311432
%N A206360 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having no more than two equal edges, and new values 0..3 introduced in row major order
%C A206360 Diagonal of A206366
%e A206360 Some solutions for n=4
%e A206360 ..0..0..1..2..3....0..0..0..0..1....0..1..0..0..2....0..0..1..0..0
%e A206360 ..0..1..2..0..0....1..2..3..1..2....2..1..2..1..1....0..1..0..1..1
%e A206360 ..1..3..0..2..0....3..2..0..1..1....3..2..0..3..3....2..0..2..2..3
%e A206360 ..3..0..2..0..3....2..3..0..2..3....0..1..3..2..1....0..3..2..3..0
%e A206360 ..1..1..2..0..2....1..3..1..0..3....1..2..1..1..0....2..3..0..1..1
%K A206360 nonn,new
%O A206360 1,1
%A A206360 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206303
%S A206303 1,1,2,8,32,184,1264,9568,79232,816128,8769536,101867776,1322831872,
%T A206303 18122579968,268425347072,4436611211264,73309336469504,
%U A206303 1303024044310528,25235367455752192,497968598916333568,10431118327503650816,234674470003955204096,5359992446798535852032
%N A206303 E.g.f.: Product_{n>=1} (1 - x^(2*n-1))^(-1/(2*n-1)).
%F A206303 Euler transform of [1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ...].
%F A206303 E.g.f.: A(x) = B(x) / sqrt(B(x^2)), where B(x) = e.g.f. of A028342.
%F A206303 E.g.f. A(x) satisfies: Product_{n>=0} A(x^(2^n))^(1/2^n) = e.g.f. of A028342.
%e A206303 G.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 32*x^4/4! + 184*x^5/5! +...
%e A206303 The e.g.f. equals the product:
%e A206303 A(x) = (1-x)^(-1) * (1-x^3)^(-1/3) * (1-x^5)^(-1/5) * (1-x^7)^(-1/7) * (1-x^9)^(-1/9) * (1-x^11)^(-1/11) *...
%o A206303 (PAR) {a(n)=n!*polcoeff(prod(m=1,n,(1-x^(2*m-1)+x*O(x^n))^(-1/(2*m-1))),n)}
%o A206303 for(n=0,31,print1(a(n),", "))
%Y A206303 Cf. A028342.
%K A206303 nonn,new
%O A206303 0,3
%A A206303 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2012

%I A206348
%S A206348 13,206,438,133,191,513,37,266,12,130,449,639,178,1028,207,397,14,13,
%T A206348 517,2060,108,898,1946,135,1225,1337,42,7717,323,144,178,542,486,64,
%U A206348 4343,593,103,4823,1110,106,1910,54,739,1238,153,8437,2087,1224,637,128
%N A206348 Smallest number k such that the decimal representations of k^2 and (k+n)^2 are anagrams.
%e A206348 a(5)=191 because 191^2 = 36481 and (191+5)^2 = 38416 and 36481 and 38416 are anagrams.
%t A206348 Table[k = 1; While[Sort[IntegerDigits[k^2]] != Sort[IntegerDigits[(k + n)^2]], k++]; k, {n, 100}] (* T. D. Noe, Feb 06 2012 *)
%K A206348 nonn,base,new
%O A206348 1,1
%A A206348 Claudio Meller (claudiomeller(AT)gmail.com), Feb 06 2012

%I A206026
%S A206026 1,12,24,72,72,168,240,336,360,504,576,720,720,720,720,1440,1440,1440,
%T A206026 1440,1440,1440,2880,2880,2880,2880,2880,2880,2880,2880,4320,4320,
%U A206026 4320,4320,4320,4320,5760,5760,8640,8640,8640,8640,8640,8640,8640,8640,8640,8640
%N A206026 a(n) = smallest number m such that sigma(k) = m has at least n positive solutions k.
%C A206026 Sequence of numbers from A145899.
%e A206026 a(6) = 168 because 168 is the smallest value of sigma(k) for n=6 positive integers k such that sigma(k) = 168 has solution; k = 60, 78, 92, 123, 143, 167.
%Y A206026 Cf. A085790, A000203, A145899, A206027.
%K A206026 nonn,new
%O A206026 1,2
%A A206026 Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Feb 03 2012

%I A206359
%S A206359 1,1,1,1,4,1,2,17,17,2,5,120,230,120,5,14,884,3284,3284,884,14,41,
%T A206359 6558,47060,91518,47060,6558,41,122,48680,674564,2545230,2545230,
%U A206359 674564,48680,122,365,361422,9669452,70894196,137375464,70894196,9669452,361422,365
%N A206359 T(n,k)=Number of nXk 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206359 Table starts
%C A206359 ...1......1.........1...........2..............5.............14.............41
%C A206359 ...1......4........17.........120............884...........6558..........48680
%C A206359 ...1.....17.......230........3284..........47060.........674564........9669452
%C A206359 ...2....120......3284.......91518........2545230.......70894196.....1973960628
%C A206359 ...5....884.....47060.....2545230......137375464.....7427049378...401471308746
%C A206359 ..14...6558....674564....70894196.....7427049378...780158903978.81929751806834
%C A206359 ..41..48680...9669452..1973960628...401471308746.81929751806834
%C A206359 .122.361422.138605744.54973461294.21705874723550
%H A206359 R. H. Hardin, <a href="/A206359/b206359.txt">Table of n, a(n) for n = 1..84</a>
%e A206359 Some solutions for n=4 k=3
%e A206359 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206359 ..3..4..1....1..0..3....3..4..0....1..0..3....1..3..4....2..0..3....2..3..1
%e A206359 ..2..3..0....2..4..0....1..2..3....2..3..4....2..4..0....1..4..0....1..4..0
%e A206359 ..4..0..2....4..1..2....2..1..4....3..2..0....0..1..2....3..1..2....3..0..2
%Y A206359 Column 1 is A007051(n-3)
%K A206359 nonn,tabl,new
%O A206359 1,5
%A A206359 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206358
%S A206358 41,48680,9669452,1973960628,401471308746,81929751806834
%N A206358 Number of nX7 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206358 Column 7 of A206359
%e A206358 Some solutions for n=4
%e A206358 ..0..1..2..0..3..4..2....0..1..2..0..3..1..2....0..1..2..3..4..0..1
%e A206358 ..2..4..1..2..4..3..0....1..3..4..1..2..4..0....4..2..0..1..2..3..4
%e A206358 ..4..2..0..1..2..0..3....2..4..3..2..0..3..1....1..0..3..4..0..1..2
%e A206358 ..1..3..4..0..1..4..2....0..1..2..3..4..0..2....3..4..1..2..3..4..0
%K A206358 nonn,new
%O A206358 1,1
%A A206358 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206357
%S A206357 14,6558,674564,70894196,7427049378,780158903978,81929751806834,
%T A206357 8606657653729993,904135652173683796,94983623219270324898,
%U A206357 9978545546504215641460,1048306488327113159222072
%N A206357 Number of nX6 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206357 Column 6 of A206359
%e A206357 Some solutions for n=4
%e A206357 ..0..1..2..0..1..2....0..1..2..0..1..2....0..1..2..0..1..2....0..1..2..0..3..4
%e A206357 ..3..2..4..3..0..4....1..0..3..2..4..1....2..0..3..2..4..0....2..4..1..2..4..0
%e A206357 ..2..0..3..1..4..0....2..3..1..4..3..0....1..2..0..3..2..4....4..0..3..4..0..1
%e A206357 ..4..3..0..4..1..2....3..2..0..1..2..3....3..4..2..0..1..2....0..1..4..3..2..4
%K A206357 nonn,new
%O A206357 1,1
%A A206357 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206356
%S A206356 5,884,47060,2545230,137375464,7427049378,401471308746,21705874723550,
%T A206356 1173549933694952,63450684804324134,3430620268033662902,
%U A206356 185485744651944072052,10028795903166889447818,542234705334673279493520
%N A206356 Number of nX5 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206356 Column 5 of A206359
%H A206356 R. H. Hardin, <a href="/A206356/b206356.txt">Table of n, a(n) for n = 1..30</a>
%e A206356 Some solutions for n=4
%e A206356 ..0..1..2..3..4....0..1..2..3..0....0..1..2..0..3....0..1..2..3..4
%e A206356 ..2..3..4..1..2....4..3..0..4..1....4..3..0..4..1....1..2..4..0..1
%e A206356 ..3..4..0..2..3....1..0..3..2..4....1..4..3..1..0....2..3..0..1..2
%e A206356 ..0..1..3..4..0....2..1..4..3..2....0..1..2..3..4....3..0..1..2..4
%K A206356 nonn,new
%O A206356 1,1
%A A206356 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206355
%S A206355 2,120,3284,91518,2545230,70894196,1973960628,54973461294,
%T A206355 1530909659694,42634040850720,1187302679649470,33064936696401300,
%U A206355 920817809589881572,25643653148853690776,714144422751974649034,19888051043349007713954
%N A206355 Number of nX4 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206355 Column 4 of A206359
%H A206355 R. H. Hardin, <a href="/A206355/b206355.txt">Table of n, a(n) for n = 1..108</a>
%F A206355 Empirical: a(n) = 32*a(n-1) +76*a(n-2) -5904*a(n-3) +1721*a(n-4) +419123*a(n-5) -145197*a(n-6) -17668949*a(n-7) +20880296*a(n-8) +307017229*a(n-9) -259902590*a(n-10) -2211508824*a(n-11) -3884842364*a(n-12) +1052899979*a(n-13) +85838941255*a(n-14) +121001157362*a(n-15) -385025973349*a(n-16) -1252681827229*a(n-17) -1873575838949*a(n-18) +4681783162473*a(n-19) +22323753578645*a(n-20) +9811253613783*a(n-21) -50670876095585*a(n-22) -100273865134560*a(n-23) -84704041294374*a(n-24) +132299239770787*a(n-25) +372399486830938*a(n-26) +189639080959232*a(n-27) -164338224373849*a(n-28) -344164045191618*a(n-29) -220338091824960*a(n-30) -45873259303120*a(n-31) +44287016659527*a(n-32) +292067697148194*a(n-33) +126193726646364*a(n-34) -159628839066498*a(n-35) -61818190307576*a(n-36) +39629245365500*a(n-37) +10915954711700*a(n-38) -10133626522160*a(n-39) +649024333200*a(n-40) +371869896960*a(n-41) -16637166720*a(n-42) for n>43
%e A206355 Some solutions for n=4
%e A206355 ..0..1..2..3....0..1..2..3....0..1..2..0....0..1..2..0....0..1..2..3
%e A206355 ..3..2..1..4....1..3..4..0....1..2..0..3....2..3..4..1....2..4..1..2
%e A206355 ..4..3..0..1....3..2..0..4....4..3..1..2....3..0..1..2....3..2..4..0
%e A206355 ..0..4..2..3....0..4..1..2....2..4..3..0....0..1..3..0....4..0..2..3
%K A206355 nonn,new
%O A206355 1,1
%A A206355 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206354
%S A206354 1,17,230,3284,47060,674564,9669452,138605744,1986829652,28480003748,
%T A206354 408243662612,5851926480128,83883833763260,1202424123191348,
%U A206354 17236024000927580,247068000076467824,3541570646368426820
%N A206354 Number of nX3 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206354 Column 3 of A206359
%H A206354 R. H. Hardin, <a href="/A206354/b206354.txt">Table of n, a(n) for n = 1..210</a>
%F A206354 Empirical: a(n) = 16*a(n-1) -20*a(n-2) -58*a(n-3) +33*a(n-4) +30*a(n-5) for n>6
%e A206354 Some solutions for n=4
%e A206354 ..0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2....0..1..2
%e A206354 ..2..0..1....1..0..3....1..0..3....3..2..4....1..3..4....1..2..3....2..3..1
%e A206354 ..1..2..3....2..3..4....2..4..0....4..3..1....2..4..3....2..0..4....1..4..0
%e A206354 ..3..4..2....3..1..2....4..1..2....2..4..3....3..0..1....3..1..2....3..0..2
%K A206354 nonn,new
%O A206354 1,2
%A A206354 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206353
%S A206353 1,4,17,120,884,6558,48680,361422,2683328,19922214,147910952,
%T A206353 1098154590,8153170064,60532632918,449420225720,3336688521102,
%U A206353 24773006668448,183925426722054,1365541253642312,10138363949208510,75271562311604144
%N A206353 Number of nX2 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206353 Column 2 of A206359
%H A206353 R. H. Hardin, <a href="/A206353/b206353.txt">Table of n, a(n) for n = 1..210</a>
%F A206353 Empirical: a(n) = 6*a(n-1) +13*a(n-2) -18*a(n-3) for n>5
%e A206353 Some solutions for n=4
%e A206353 ..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1
%e A206353 ..1..0....1..2....2..3....1..0....1..2....1..0....2..3....1..2....2..3....1..0
%e A206353 ..2..3....2..0....3..2....2..3....3..4....2..3....1..4....2..3....4..2....2..3
%e A206353 ..3..4....3..4....4..0....4..1....2..3....3..1....0..1....3..0....0..1....0..4
%K A206353 nonn,new
%O A206353 1,2
%A A206353 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206352
%S A206352 1,4,230,91518,137375464,780158903978
%N A206352 Number of nXn 0..4 arrays with no element equal to another within two positions in the same row or column, and new values 0..4 introduced in row major order
%C A206352 Diagonal of A206359
%e A206352 Some solutions for n=4
%e A206352 ..0..1..2..0....0..1..2..3....0..1..2..0....0..1..2..0....0..1..2..3
%e A206352 ..1..0..3..1....2..0..1..4....1..2..0..3....2..0..3..1....1..3..4..0
%e A206352 ..4..2..0..3....3..4..0..2....4..3..1..2....4..3..1..4....3..2..0..4
%e A206352 ..2..1..4..0....0..1..3..0....2..4..3..0....0..4..2..0....0..4..1..2
%K A206352 nonn,new
%O A206352 1,2
%A A206352 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206328
%S A206328 5,17,197,577,2917,15377,41617,147457,215297,401957,414737,509797,
%T A206328 1196837,1308737,1378277,1547537,1623077,1726597,1887877,2446097,
%U A206328 2604997,2802277,2835857,3857297,4218917,4343057,4384837,5779217,6022117,6421157,7096897,8031557
%N A206328 Primes of the form n^2+1 such that (n+2)^2+1 is also prime.
%C A206328 Primes corresponding to A096012 and subset of A002496.
%C A206328 For n > 1, a(n) ==7 (mod 10) because n ==4 (mod 10).
%C A206328 Conjecture: this sequence is infinite.
%e A206328 For n = 4, n^2 + 1 = 17 is prime and  (n+2)^2 + 1 = 37 is also prime => 17 is in the sequence.
%p A206328 for n from 1 to 4000 do: x:=n^2+1:y:=(n+2)^2+1:if type(x,prime)=true and type(y,prime)=true then printf(`%d, `,x): else fi:od:
%Y A206328 Cf. A096012, A002496.
%K A206328 nonn,easy,new
%O A206328 1,1
%A A206328 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 06 2012

%I A206346
%S A206346 0,1,1,4,8,72,236,3665,19037
%N A206346 Number of solvable clock puzzles with n positions in Final Fantasy XIII-2, up to rotation and reflection.
%C A206346 Equals the number of Hamiltonian directed graphs on n vertices with the properties that: (1) every vertex has outdegree 1 or 2; and (2) the vertices can be arranged in a circle so that the directed edges leaving each vertex are symmetric about that vertex (e.g., if there is a directed edge that points two vertices in the clockwise direction, then the other one must point two vertices in the counter-clockwise direction).
%C A206346 The same as A206345, except clock puzzles that are simply rotations or reflections of each other are not counted multiple times.
%H A206346 N. Johnston, <a href="http://www.njohnston.ca/2012/02/counting-and-solving-final-fantasy-xiii-2s-clock-puzzles/">Counting and Solving Final Fantasy XIII-2's Clock Puzzles</a>
%Y A206346 Cf. A206344, A206345.
%K A206346 nonn,more,new
%O A206346 1,4
%A A206346 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 06 2012

%I A206345
%S A206345 0,1,1,13,32,507,1998,33136,193995,3426518
%N A206345 Number of solvable clock puzzles with n positions in Final Fantasy XIII-2.
%C A206345 The sequence gives the number of ways of placing the integers 1, 2, ..., floor(n/2) (with repetition) in n spaces on a circle so that you can jump to every integer exactly once, and the distance you jump is equal to the integer you are currently standing on.
%C A206345 A206344 is a trivial upper bound.
%C A206345 This is the same as A206346, except clock puzzles that are rotations or reflections of each other are counted as distinct.
%H A206345 N. Johnston, <a href="http://www.njohnston.ca/2012/02/counting-and-solving-final-fantasy-xiii-2s-clock-puzzles/">Counting and Solving Final Fantasy XIII-2's Clock Puzzles</a>
%e A206345 A solvable clock puzzle in the n = 6 case arises from the following integers (placed clockwise around a circle): 1, 3, 3, 2, 1, 3. If we label the positions 0, 1, 2, 3, 4, 5, then a solution to this puzzle is the following sequence of positions: 0, 1, 4, 3, 5, 2.
%Y A206345 Cf. A206344, A206346.
%K A206345 nonn,more,new
%O A206345 1,4
%A A206345 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 06 2012
%E A206345 a(10) from Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 07 2012

%I A206344
%S A206344 0,1,1,16,32,729,2187,65536,262144,9765625,48828125,2176782336,
%T A206344 13060694016,678223072849,4747561509943,281474976710656,
%U A206344 2251799813685248,150094635296999121,1350851717672992089,100000000000000000000,1000000000000000000000
%N A206344 Floor(n/2)^n.
%C A206344 The sequence gives the number of (potentially unsolvable) "clock puzzles" with n positions in the video game Final Fantasy XIII-2.
%H A206344 N. Johnston, <a href="http://www.njohnston.ca/2012/02/counting-and-solving-final-fantasy-xiii-2s-clock-puzzles/">Counting and Solving Final Fantasy XIII-2's Clock Puzzles</a>
%p A206344 seq(floor(n/2)^n, n=1..50);
%Y A206344 Cf. A206345, A206346.
%K A206344 nonn,easy,new
%O A206344 1,4
%A A206344 Nathaniel Johnston (nathaniel(AT)njohnston.ca), Feb 06 2012

%I A194801
%S A194801 0,0,1,0,1,0,0,1,1,3,0,1,2,4,1,0,1,3,5,4,6,0,1,4,6,7,9,3,0,1,5,7,10,
%T A194801 12,9,10,0,1,6,8,13,15,15,16,6,0,1,7,9,16,18,21,22,16,15,0,1,8,10,19,
%U A194801 21,27,28,26,25,10,0,1,9,11,22,24,33,34,36,35
%N A194801 Square array read by antidiagonals: T(n,k) = k*((n+1)*k-n+1)/2, k = 0, +- 1, +- 2,..., n >= 0.
%C A194801 Note that a unique formula gives several types of numbers. Row 0 lists 0 together the Molien series for 3-dimensional group [2,k]+ = 22k. Row 1 lists, except first zero, the squares repeated. If n >= 2, row n lists the generalized (n+3)-gonal numbers, for example: row 2 list the generalized pentagonal numbers A001318. See some other examples in the cross-references section.
%e A194801 Array begins:
%e A194801 (A008795): 0, 1,  0,  3,  1,  6,  3, 10,   6,  15,  10...
%e A194801 (A008794): 0, 1,  1,  4,  4,  9,  9, 16,  16,  25,  25...
%e A194801 A001318:   0, 1,  2,  5,  7, 12, 15, 22,  26,  35,  40...
%e A194801 A000217:   0, 1,  3,  6, 10, 15, 21, 28,  36,  45,  55...
%e A194801 A085787:   0, 1,  4,  7, 13, 18, 27, 34,  46,  55,  70...
%e A194801 A001082:   0, 1,  5,  8, 16, 21, 33, 40,  56,  65,  85...
%e A194801 A118277:   0, 1,  6,  9, 19, 24, 39, 46,  66,  75, 100...
%e A194801 A074377:   0, 1,  7, 10, 22, 27, 45, 52,  76,  85, 115...
%e A194801 A195160:   0, 1,  8, 11, 25, 30, 51, 58,  86,  95, 130...
%e A194801 A195162:   0, 1,  9, 12, 28, 33, 57, 64,  96, 105, 145...
%e A194801 A195313:   0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
%e A194801 A195818:   0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...
%o A194801 GWBASIC
%o A194801 100 'SQUARE ARRAY T(N,K) A194801
%o A194801 200 DIM T(11,10)
%o A194801 210 FOR N=0 TO 11
%o A194801 220  FOR J=0 TO 5
%o A194801 230    IF J>0 THEN T(N,K)=  J *((N+1)*  J -N+1)/2: K=K+1
%o A194801 240                T(N,K)=(-J)*((N+1)*(-J)-N+1)/2: K=K+1
%o A194801 250  NEXT J
%o A194801 260  K=0
%o A194801 270 NEXT N
%o A194801 299 '...
%o A194801 300 'PRINT SQUARE ARRAY T(N,K) (see example)
%o A194801 310 FOR N=0 TO 11
%o A194801 320  FOR K=0 TO 10
%o A194801 330   PRINT T(N,K);
%o A194801 340  NEXT K
%o A194801 350  PRINT
%o A194801 360 NEXT N
%o A194801 500 END
%Y A194801 Rows (0-11): 0 together with A008795, (truncated A008794), A001318, A000217, A085787, A001082, A118277, A074377, A195160, A195162, A195313, A195818
%Y A194801 Columns (0-9): A000004, A000012, A001477, (truncated A000027), A016777, (truncated A008585), A016945, (truncated A016957), A017341, (truncated A017329).
%Y A194801 Cf. A139600.
%K A194801 nonn,tabl,new
%O A194801 0,10
%A A194801 Omar E. Pol (info(AT)polprimos.com), Feb 05 2012

%I A206245
%S A206245 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,
%T A206245 4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,
%U A206245 7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8
%N A206245 Number of partitions of n into repunit powers, cf. A083278.
%C A206245 a(n) = A206244(n) for n <= 120, a(n) > A206244(n) for n > 120.
%H A206245 Reinhard Zumkeller, <a href="/A206245/b206245.txt">Table of n, a(n) for n = 0..1000</a>
%H A206245 Wikipedia, <a href="http://en.wikipedia.org/wiki/Repunit">Repunit</a>
%H A206245 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H A206245 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%o A206245 (Haskell)
%o A206245 a206245 = p a083278_list where
%o A206245    p _      0 = 1
%o A206245    p rps'@(rp:rps) n = if n < rp then 0 else p rps' (n - rp) + p rps n
%Y A206245 Cf. A002275, A000041, A179051.
%K A206245 nonn,new
%O A206245 0,12
%A A206245 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 05 2012

%I A206244
%S A206244 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,
%T A206244 4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,
%U A206244 7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8
%N A206244 Number of partitions of n into repunits, cf. A002275.
%C A206244 a(n) = A206245(n) for n <= 120, a(n) < A206245(n) for n > 120.
%H A206244 Reinhard Zumkeller, <a href="/A206244/b206244.txt">Table of n, a(n) for n = 0..1000</a>
%H A206244 Wikipedia, <a href="http://en.wikipedia.org/wiki/Repunit">Repunit</a>
%H A206244 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Repunit.html">Repunit</a>
%H A206244 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%o A206244 (Haskell)
%o A206244 a206244 = p $ tail a002275_list where
%o A206244    p _             0 = 1
%o A206244    p rus'@(ru:rus) n = if n < ru then 0 else p rus' (n - ru) + p rus n
%Y A206244 Cf. A000041, A179051.
%K A206244 nonn,new
%O A206244 0,12
%A A206244 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 05 2012

%I A206347
%S A206347 21,27,33,60,117,153,222,228,306,426,480,495,558,585,615,636,669,684,
%T A206347 762,768,819,852,894,909,1083,1125,1131,1224,1239,1341,1455,1512,1539,
%U A206347 1776,1812,1845,2301,2484,2517,2541,2604,2706,2769,3093,3177
%N A206347 n such that 10*n+1, 20*n+1, and 30*n+1 are all primes.
%C A206347 (10*n+1)*(20*n+1)*(30*n+1) is a Carmichael number for all n in this sequence. Why is (6m+1)*(12m+1)*(18m +1) used to generate Carmichael numbers and never the formula (10m+1)*(20m+1)*(30m+1)?
%t A206347 Select[Range[20000], PrimeQ[10 #+1] && PrimeQ[20 #+1] && PrimeQ[30 #+1]&]
%o A206347 Cf. A002997, A046025.
%K A206347 nonn,new
%O A206347 1,1
%A A206347 José María Grau Ribas (grau.ribas(AT)gmail.com), Feb 06 2012

%I A185023
%S A185023 0,2,1,5,4,3,3,7,2,21,6,6,20,10,5,5,19,4,9,4,4,23,18,8,8,8,3,27,22,22,
%T A185023 17,12,7,7,7,7,26,21,21,16,16,11,11,11,6,6,6,6,25,20,20,20,15,5,15,10,
%U A185023 10,10,10,5,5,29,5,24,24,24,19,19,19,19,14,9,14,9
%N A185023 Number of steps to reach the number 1 of the map  n -> n/3 if x==0 (mod 3), n -> (4n-1)/3 if x==1 (mod 3), n -> (4n+1)/3 if x==2 (mod 3).
%C A185023 A variation of the "3x+1" problem. - T. D. Noe, Feb 06 2012
%H A185023 Michel Lagneau, <a href="/A185023/b185023.txt">Table of n, a(n) for n = 1..10000</a>
%e A185023 a(8) = 7 because the trajectory of 8  is  8 -> 11 -> 15 -> 5 -> 7-> 9 -> 3-> 1 with 7 iterations.
%p A185023 T:=array(0..2):for n from 1 to 100 do:ii:=0:n0:=n: i:=0:for it from 1 to 100 while(ii=0) do:T[0]:=n0/3: T[1]:=(4*n0-1)/3: T[2]:=(4*n0+1)/3:r:=irem(n0,3):n0:=T[r]: i:=i+1:if n0=1 then ii:=1: else fi:od: printf(`%d, `,i):od:
%Y A185023 Cf. A006577 (3x+1 problem).
%K A185023 nonn,new
%O A185023 1,2
%A A185023 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 04 2012

%I A206343
%S A206343 192,96,96,455,181,455,296,232,232,296,1376,534,1340,534,1376,880,686,
%T A206343 682,682,686,880,4111,1589,4007,1581,4007,1589,4111,2622,2042,2030,
%U A206343 2034,2034,2030,2042,2622,12319,4751,12005,4711,12021,4711,12005,4751,12319
%N A206343 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206343 Table starts
%C A206343 ..192...96...455...296..1376...880...4111...2622..12319...7854..36945...23550
%C A206343 ...96..181...232...534...686..1589...2042...4751...6110..14239..18318...42705
%C A206343 ..455..232..1340...682..4007..2030..12005...6074..35999..18206.107983...54606
%C A206343 ..296..534...682..1581..2034..4711...6078..14101..18210..42271..54606..126799
%C A206343 .1376..686..4007..2034.12021..6090..36015..18222.107997..54618.323943..163806
%C A206343 ..880.1589..2030..4711..6090.14181..18258..42351..54654.126861.163842..380391
%C A206343 .4111.2042.12005..6078.36015.18258.108141..54762.324087.163950.971925..491514
%C A206343 .2622.4751..6074.14101.18222.42351..54762.127581.164274.381111.491838.1141701
%H A206343 R. H. Hardin, <a href="/A206343/b206343.txt">Table of n, a(n) for n = 1..2380</a>
%F A206343 Empirical for column k:
%F A206343 k=1: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>10
%F A206343 k=2: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>11
%F A206343 k=3: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>12
%F A206343 k=4: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>13
%F A206343 k=5: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>14
%F A206343 k=6: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>15
%F A206343 k=7: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>16
%F A206343 apparently a(n-1)+3*a(n-2)-3*a(n-3) for n>k+9
%e A206343 Some solutions for n=4 k=3
%e A206343 ..3..1..3..1....3..2..3..2....0..2..0..1....1..3..0..1....1..1..0..2
%e A206343 ..1..3..1..3....2..3..2..3....2..0..2..0....1..0..3..0....3..0..1..0
%e A206343 ..3..1..3..1....3..2..3..2....0..2..0..2....0..1..0..3....0..3..0..1
%e A206343 ..1..3..1..3....2..3..2..3....1..0..2..0....1..0..1..0....2..0..3..0
%e A206343 ..3..1..3..1....3..2..3..2....0..1..0..2....0..1..0..1....0..2..0..3
%K A206343 nonn,tabl,new
%O A206343 1,1
%A A206343 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206342
%S A206342 4111,2042,12005,6078,36015,18258,108141,54762,324087,163950,971925,
%T A206342 491514,2915439,1474206,8746143,4422606,26238417,13267806,78715239,
%U A206342 39803406,236145705,119410206,708437103,358230606,2125311297,1074691806
%N A206342 Number of (n+1)X8 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206342 Column 7 of A206343
%H A206342 R. H. Hardin, <a href="/A206342/b206342.txt">Table of n, a(n) for n = 1..210</a>
%F A206342 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>16
%e A206342 Some solutions for n=4
%e A206342 ..1..2..0..2..0..2..0..1....3..3..0..1..0..1..0..1....1..3..0..1..0..3..0..1
%e A206342 ..1..0..2..0..2..0..2..0....3..0..3..0..1..0..1..0....3..0..3..0..1..0..3..0
%e A206342 ..0..1..0..2..0..2..0..2....0..3..0..3..0..1..0..1....0..3..0..3..0..1..0..3
%e A206342 ..1..0..1..0..2..0..2..0....1..0..3..0..3..0..1..0....3..0..3..0..3..0..1..0
%e A206342 ..0..1..0..1..0..2..0..2....0..1..0..3..0..3..0..1....0..3..0..3..0..3..0..1
%K A206342 nonn,new
%O A206342 1,1
%A A206342 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206341
%S A206341 880,1589,2030,4711,6090,14181,18258,42351,54654,126861,163842,380391,
%T A206341 491406,1141143,1474206,3423417,4422606,10270239,13267806,30810705,
%U A206341 39803406,92432103,119410206,277296297,358230606,831888879,1074691806
%N A206341 Number of (n+1)X7 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206341 Column 6 of A206343
%H A206341 R. H. Hardin, <a href="/A206341/b206341.txt">Table of n, a(n) for n = 1..210</a>
%F A206341 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>15
%e A206341 Some solutions for n=4
%e A206341 ..0..1..0..1..0..2..0....1..0..1..0..1..0..1....3..3..0..2..0..2..0
%e A206341 ..3..0..1..0..1..0..2....0..1..0..1..0..1..0....2..0..3..0..2..0..2
%e A206341 ..0..3..0..1..0..1..0....1..0..1..0..1..0..1....0..2..0..3..0..2..0
%e A206341 ..1..0..3..0..1..0..1....0..1..0..1..0..1..0....1..0..2..0..3..0..2
%e A206341 ..0..1..0..3..0..1..1....1..0..1..0..1..0..1....0..1..0..2..0..3..1
%K A206341 nonn,new
%O A206341 1,1
%A A206341 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206340
%S A206340 1376,686,4007,2034,12021,6090,36015,18222,107997,54618,323943,163806,
%T A206340 971799,491406,2915385,1474206,8746143,4422606,26238417,13267806,
%U A206340 78715239,39803406,236145705,119410206,708437103,358230606,2125311297
%N A206340 Number of (n+1)X6 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206340 Column 5 of A206343
%H A206340 R. H. Hardin, <a href="/A206340/b206340.txt">Table of n, a(n) for n = 1..210</a>
%F A206340 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>14
%e A206340 Some solutions for n=4
%e A206340 ..0..1..0..2..0..2....1..2..1..2..1..2....1..1..0..2..0..1....1..0..1..0..2..0
%e A206340 ..1..0..1..0..2..0....2..1..2..1..2..1....2..0..1..0..2..0....0..1..0..1..0..2
%e A206340 ..0..1..0..1..0..2....1..2..1..2..1..2....0..2..0..1..0..2....1..0..1..0..1..0
%e A206340 ..3..0..1..0..1..0....2..1..2..1..2..1....2..0..2..0..1..0....0..1..0..1..0..1
%e A206340 ..0..3..0..1..0..1....1..2..1..2..1..2....0..2..0..2..0..1....3..0..1..0..1..2
%K A206340 nonn,new
%O A206340 1,1
%A A206340 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206339
%S A206339 296,534,682,1581,2034,4711,6078,14101,18210,42271,54606,126799,
%T A206339 163806,380385,491406,1141143,1474206,3423417,4422606,10270239,
%U A206339 13267806,30810705,39803406,92432103,119410206,277296297,358230606,831888879
%N A206339 Number of (n+1)X5 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206339 Column 4 of A206343
%H A206339 R. H. Hardin, <a href="/A206339/b206339.txt">Table of n, a(n) for n = 1..210</a>
%F A206339 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>13
%e A206339 Some solutions for n=4
%e A206339 ..3..1..3..1..3....1..3..1..3..1....0..1..0..1..0....1..0..1..0..2
%e A206339 ..1..3..1..3..1....3..1..3..1..3....1..0..1..0..1....0..1..0..1..0
%e A206339 ..3..1..3..1..3....1..3..1..3..1....0..1..0..1..0....2..0..1..0..1
%e A206339 ..1..3..1..3..1....3..1..3..1..3....3..0..1..0..1....0..2..0..1..0
%e A206339 ..3..1..3..1..3....1..3..1..3..1....0..3..0..1..3....2..0..2..0..1
%K A206339 nonn,new
%O A206339 1,1
%A A206339 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206338
%S A206338 455,232,1340,682,4007,2030,12005,6074,35999,18206,107983,54606,
%T A206338 323937,163806,971799,491406,2915385,1474206,8746143,4422606,26238417,
%U A206338 13267806,78715239,39803406,236145705,119410206,708437103,358230606,2125311297
%N A206338 Number of (n+1)X4 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206338 Column 3 of A206343
%H A206338 R. H. Hardin, <a href="/A206338/b206338.txt">Table of n, a(n) for n = 1..210</a>
%F A206338 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>12
%e A206338 Some solutions for n=4
%e A206338 ..2..2..0..1....0..2..0..1....0..2..0..1....1..2..0..2....0..1..0..1
%e A206338 ..1..0..2..0....3..0..2..0....2..0..2..0....1..0..2..0....1..0..1..0
%e A206338 ..0..1..0..2....0..3..0..2....0..2..0..2....0..1..0..2....0..1..0..1
%e A206338 ..2..0..1..0....3..0..3..0....1..0..2..0....1..0..1..0....2..0..1..0
%e A206338 ..0..2..0..1....0..3..0..3....0..1..0..2....0..1..0..1....0..2..0..1
%K A206338 nonn,new
%O A206338 1,1
%A A206338 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206337
%S A206337 96,181,232,534,686,1589,2042,4751,6110,14239,18318,42705,54942,
%T A206337 128103,164814,384297,494430,1152879,1483278,3458625,4449822,10375863,
%U A206337 13349454,31127577,40048350,93382719,120145038,280148145,360435102,840444423
%N A206337 Number of (n+1)X3 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206337 Column 2 of A206343
%H A206337 R. H. Hardin, <a href="/A206337/b206337.txt">Table of n, a(n) for n = 1..210</a>
%F A206337 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>11
%e A206337 Some solutions for n=4
%e A206337 ..3..0..1....0..1..0....1..3..1....0..2..0....0..1..0....1..0..1....0..1..0
%e A206337 ..0..3..0....3..0..1....3..1..3....3..0..2....1..0..1....0..1..0....3..0..1
%e A206337 ..2..0..3....0..3..0....1..3..1....0..3..0....0..1..0....2..0..1....0..3..0
%e A206337 ..0..2..0....1..0..3....3..1..3....1..0..3....2..0..1....0..2..0....3..0..3
%e A206337 ..1..0..2....0..1..0....1..3..1....0..1..3....0..2..1....1..0..2....0..3..3
%K A206337 nonn,new
%O A206337 1,1
%A A206337 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206336
%S A206336 192,96,455,296,1376,880,4111,2622,12319,7854,36945,23550,110823,
%T A206336 70638,332457,211902,997359,635694,2992065,1907070,8976183,5721198,
%U A206336 26928537,17163582,80785599,51490734,242356785,154472190,727070343,463416558
%N A206336 Number of (n+1)X2 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206336 Column 1 of A206343
%H A206336 R. H. Hardin, <a href="/A206336/b206336.txt">Table of n, a(n) for n = 1..210</a>
%F A206336 Empirical: a(n) = a(n-1) +3*a(n-2) -3*a(n-3) for n>10
%e A206336 Some solutions for n=4
%e A206336 ..1..1....2..1....1..3....0..3....3..0....3..0....0..1....0..1....1..0....1..1
%e A206336 ..3..0....3..0....2..0....3..0....0..3....0..3....2..0....1..0....0..1....1..0
%e A206336 ..0..3....0..3....0..2....0..3....2..0....3..0....0..2....0..1....2..0....0..1
%e A206336 ..1..0....3..0....1..0....2..0....0..2....0..3....2..0....1..0....0..2....3..0
%e A206336 ..0..1....0..3....0..1....0..2....1..0....1..1....0..2....0..1....3..1....0..3
%K A206336 nonn,new
%O A206336 1,1
%A A206336 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206335
%S A206335 192,181,1340,1581,12021,14181,108141,127581,973221,1148181,8758941,
%T A206335 10333581,78830421,93002181,709473741,837019581,6385263621,7533176181,
%U A206335 57467372541,67798585581,517206352821,610187270181,4654857175341
%N A206335 Number of (n+1)X(n+1) 0..3 arrays with every 2X2 subblock having nonzero determinant and commuting with every horizontal or vertical neighbor
%C A206335 Diagonal of A206343
%H A206335 R. H. Hardin, <a href="/A206335/b206335.txt">Table of n, a(n) for n = 1..34</a>
%e A206335 Some solutions for n=4
%e A206335 ..1..0..2..0..2....0..1..0..1..0....1..2..0..2..0....3..3..0..1..0
%e A206335 ..0..1..0..2..0....1..0..1..0..1....2..0..2..0..2....1..0..3..0..1
%e A206335 ..1..0..1..0..2....0..1..0..1..0....0..2..0..2..0....0..1..0..3..0
%e A206335 ..0..1..0..1..0....3..0..1..0..1....1..0..2..0..2....1..0..1..0..3
%e A206335 ..1..0..1..0..1....0..3..0..1..3....0..1..0..2..0....0..1..0..1..3
%K A206335 nonn,new
%O A206335 1,1
%A A206335 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206306
%S A206306 1,0,1,0,3,1,0,7,6,1,0,15,23,9,1,0,31,72,48,12,1,0,63,201,198,82,15,1,
%T A206306 0,127,522,699,420,125,18,1,0,255,1291,2223,1795,765,177,21,1,0,511,
%U A206306 3084,6562,6768,3840,1260,238,24,1
%N A206306 Riordan array (1, x/(1-3*x+2*x^2)).
%C A206306 Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A206306 Diagonals sums are even indexed Fibonacci numbers.
%F A206306 Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
%F A206306 G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
%F A206306 T(n,n) = 1, T(n+1,n) = 3n = A008585(n), T(n+2,n) = A062725(n).
%e A206306 Triangle begins :
%e A206306 1
%e A206306 0, 1
%e A206306 0, 3, 1
%e A206306 0, 7, 6, 1
%e A206306 0, 15, 23, 9, 1
%e A206306 0, 31, 72, 48, 12, 1
%e A206306 0, 63, 201, 198, 82, 15, 1
%e A206306 0, 127, 522, 699, 420, 125, 18, 1
%e A206306 0, 255, 1291, 2223, 1795, 765, 177, 21, 1
%e A206306 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1
%e A206306 0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1
%Y A206306 Cf. Columns : A000007, A000225 (Mersenne numbers), A045618, A055582,
%Y A206306 Cf. A110441
%K A206306 easy,nonn,tabl,new
%O A206306 0,5
%A A206306 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 06 2012

%I A185320
%S A185320 1,4,112,8608,1295104,322018816,119597651968,62037189087232,
%T A185320 42842215767801856,38001850792907505664,42106195262186260529152,
%U A185320 56992583802129636291248128,92535374062287540141093289984,177509548409832461509746497683456,397176345992538727622693418988208128
%N A185320 E.g.f. A(x)=sum(n>=0, a(n)*x^(2*n+1)/(2*n+1)! is inverse to f(x)=2*arctan(x)-x
%F A185320 a(n)=sum(k=1..2*n, (2*n+k)!*sum(j=1..k, sum(l=0..j-1, ((-1)^(n+l+j)*sum(i=j-l..2*n-l+j, (2^i*stirling1(i,j-l)*binomial(2*n-l+j-1,i-1))/i!))/l!)/(k-j)!)), n>0, a(0)=1.
%o A185320 (Maxima) a(n):=if n=0 then 1 else sum((2*n+k)!*sum(sum(((-1)^(n+l+j)*sum((2^i*stirling1(i,j-l)*binomial(2*n-l+j-1,i-1))/i!,i,j-l,2*n-l+j))/l!,l,0,j-1)/(k-j)!,j,1,k),k,1,2*n);
%K A185320 nonn,new
%O A185320 0,2
%A A185320 Vladimir Kruchinin (kru(AT)ie.tusur.ru), Feb 05 2012

%I A184990
%S A184990 1,1,0,2,1,2,4,2,2,6,4,4,10,6,8,16,9,10,24,14,16,36,20,
%T A184990 24,53,30,32,76,43,48,108,60,68,150,84,92,206,114,128,280,
%U A184990 155,172,376,208,228,504,276,304,668,366,400,878,480,524,1148
%V A184990 1,1,0,2,-1,-2,4,-2,-2,6,-4,-4,10,-6,-8,16,-9,-10,24,-14,-16,36,-20,
%W A184990 -24,53,-30,-32,76,-43,-48,108,-60,-68,150,-84,-92,206,-114,-128,280,
%X A184990 -155,-172,376,-208,-228,504,-276,-304,668,-366,-400,878,-480,-524,1148
%N A184990 McKay-Thompson series of class 24C for the Monster group with a(0) = 1.
%C A184990 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A184990 M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>
%H A184990 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A184990 Expansion of (1/q) * psi(q) * phi(-q^4) / (psi(-q^3) * psi(-q^6)) in powers of q where phi(), psi() are Rmanujan theta functions.
%F A184990 Expansion of eta(q^2)^2 * eta(q^4)^2 / (eta(q) * eta(q^3) * eta(q^8)* eta(q^24)) in powers of q.
%F A184990 Euler transform of period 24 sequence [ 1, -1, 2, -3, 1, 0, 1, -2, 2, -1, 1, -2, 1, -1, 2, -2, 1, 0, 1, -3, 2, -1, 1, 0, ...].
%F A184990 a(n) = A058573(n) unless n = 0.
%e A184990 1/q + 1 + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 + ...
%o A184990 (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^8 + A)* eta(x^24 + A)), n))}
%Y A184990 Cf. A058573.
%K A184990 sign,new
%O A184990 -1,4
%A A184990 Michael Somos, Feb 05 2012

%I A206299
%S A206299 1,1,0,2,1,2,4,2,2,6,4,4,10,6,8,16,9,10,24,14,16,36,20,
%T A206299 24,53,30,32,76,43,48,108,60,68,150,84,92,206,114,128,280,
%U A206299 155,172,376,208,228,504,276,304,668,366,400,878,480,524,1148
%V A206299 1,-1,0,2,-1,-2,4,-2,-2,6,-4,-4,10,-6,-8,16,-9,-10,24,-14,-16,36,-20,
%W A206299 -24,53,-30,-32,76,-43,-48,108,-60,-68,150,-84,-92,206,-114,-128,280,
%X A206299 -155,-172,376,-208,-228,504,-276,-304,668,-366,-400,878,-480,-524,1148
%N A206299 McKay-Thompson series of class 24C for the Monster group with a(0) = -1.
%C A206299 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A206299 M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>
%H A206299 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A206299 Expansion of (1/q) * chi(q) * chi(q^12)^3 / (chi(q^3)^3 * chi(q^4)) in powers of q where chi() is a Ramanujan theta function.
%F A206299 Expansion of c(q^2)*c(q^4)/(c(q)*c(q^8)) in powers of q where c() is a cubic AGM theta function.
%F A206299 Euler transform of period 24 sequence [ -1, 0, 2, 1, -1, 0, -1, 0, 2, 0, -1, -2, -1, 0, 2, 0, -1, 0, -1, 1, 2, 0, -1, 0, ...].
%F A206299 a(n) = A058573(n) unless n = 0.
%e A206299 1/q - 1 + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 + ...
%o A206299 (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^4 +A ) * eta(x^24 + A)^3), n))}
%Y A206299 Cf. A058573, A206298, A184990.
%K A206299 sign,new
%O A206299 -1,4
%A A206299 Michael Somos, Feb 05 2012

%I A206296
%S A206296 2,3,10,63,2750,842751,85558343750,2098355820117528699,
%T A206296 769999781728184386440152910156250,
%U A206296 2359414683424785920146467280333749864720543920418139851
%N A206296 Numbers matched to the Fibonacci polynomials.
%C A206296 The matching is given at A206284.  A guide to matched
%C A206296 pairs, (polynomial sequence, integer sequence):
%C A206296 ...
%C A206296 polynomial sequence....integer sequence
%C A206296 x^n....................A000040
%C A206296 (x+1)^n................A007188
%C A206296 n*x^(n-1)..............A062457
%C A206296 (1-x^n)/(1-x)..........A002110
%C A206296 n+(n-1)x+...+x^n.......A006939
%e A206296 Referring to the matching and example at A206284:
%e A206296 1 -> 2
%e A206296 x -> 3
%e A206296 1 + x^2 -> 10
%e A206296 2x + x^3 -> 63
%e A206296 1 + 3x^2 + x^4 -> 2750
%t A206296 c[n_] := CoefficientList[Fibonacci[n, x], x]
%t A206296 f[n_] := Product[Prime[k]^c[n][[k]],
%t A206296  {k, 1, Length[c[n]]}]
%t A206296 Table[f[n], {n, 1, 11}]  (* A206296 *)
%Y A206296 Cf. A206284.
%K A206296 nonn,new
%O A206296 1,1
%A A206296 Clark Kimberling (ck6(AT)evansville.edu), Feb 05 2012

%I A206295
%S A206295 1,15,7875,11142140625,467812950752090302734375,
%T A206295 202822649430450649753225796950422853099588897705078125
%N A206295 (1/6)*A007188(n).
%C A206295 For n>1, a(n) is divisible by the product of the first n odd primes.
%t A206295 c[n_] := CoefficientList[(1 + x)^n, x]
%t A206295 f[n_] := Product[Prime[k]^c[n][[k]],
%t A206295  {k, 1, Length[c[n]]}]
%t A206295 t=Table[f[n], {n, 1, 7}] (* A007188 *)
%t A206295 t/6                      (* A206295 *)
%Y A206295 Cf. A007188.
%K A206295 nonn,new
%O A206295 1,2
%A A206295 Clark Kimberling (ck6(AT)evansville.edu), Feb 05 2012

%I A206285
%S A206285 1,2,4,5,7,8,11,13,14,15,16,17,19,21,23,25,26,29,31,32,33,34,35,37,38,
%T A206285 39,41,43,45,47,49,51,53,55,57,58,59,61,62,63,64,65,67,69,71,73,74,75,
%U A206285 77,78,79,82,83,84,85,86,87,89,90,91,93,94,95,97,99,101,103
%N A206285 Numbers that match polynomials not irreducible over the nonnegative integers.
%C A206285 Complement of A206284.
%e A206285 (See the example at A206284.)
%t A206285 b[n_] := Table[x^k, {k, 0, n}];
%t A206285 f[n_] := f[n] = FactorInteger[n]; z = 400;
%t A206285 t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
%t A206285 == Prime[k], f[n][[m, 2]], 0];
%t A206285 u = Table[Apply[Plus,
%t A206285     Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
%t A206285       Length[f[n]]}]], {n, 1, z}];
%t A206285 p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
%t A206285 Table[p[n, x], {n, 1, z/4}]
%t A206285 v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
%t A206285 AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
%t A206285 Complement[Range[200], v]      (* A206285 *)
%Y A206285 Cf. A206285, A206331.
%K A206285 nonn,new
%O A206285 1,2
%A A206285 Clark Kimberling (ck6(AT)evansville.edu), Feb 05 2012

%I A206284
%S A206284 3,6,9,10,12,18,20,22,24,27,28,30,36,40,42,44,46,48,50,52,54,56,60,66,
%T A206284 68,70,72,76,80,81,88,92,96,98,100,102,104,108,112,114,116,118,120,
%U A206284 124,126,130,132,136,140,144,148,150,152,154,160,162,164,168,170
%N A206284 Numbers that match irreducible polynomials over the nonnegative integers.
%C A206284 Each n>1 matches a polynomial having nonnegative integer
%C A206284 coefficients determined by the prime factorization of n.
%C A206284 Write n=p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k).  The
%C A206284 matching polynomial is then
%C A206284 p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.  Identities:
%C A206284 p(m*n)=p(m,x)+p(n,x)
%C A206284 p(m*n)=p(GCD(m,n),x)+p(LCM(m,n),x)
%C A206284 p(m+n)=p(GCD(m,n),x)+p((m+n)/(GCD(m,n),x), so that if A003057 is read as a square matrix, then
%C A206284 p(A003057,x)=p(A003988,x)+p(106448,x).
%e A206284 Polynomials having nonnegative integer coefficients are
%e A206284 matched to the positive integers as follows:
%e A206284 n ... p[n,x] .. irreducible
%e A206284 1 ... 0 ....... no
%e A206284 2 ... 1 ....... no
%e A206284 3 ... x ....... yes
%e A206284 4 ... 2 ....... no
%e A206284 5 ... x^2 ..... no
%e A206284 6 ... 1+x ..... yes
%e A206284 7 ... x^3 ..... no
%e A206284 8 ... 3 ....... no
%e A206284 9 ... 2x ...... yes
%e A206284 10 .. 1+x^2 ... yes
%t A206284 b[n_] := Table[x^k, {k, 0, n}];
%t A206284 f[n_] := f[n] = FactorInteger[n]; z = 400;
%t A206284 t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
%t A206284 == Prime[k], f[n][[m, 2]], 0];
%t A206284 u = Table[Apply[Plus,
%t A206284     Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
%t A206284       Length[f[n]]}]], {n, 1, z}];
%t A206284 p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
%t A206284 Table[p[n, x], {n, 1, z/4}]
%t A206284 v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
%t A206284 AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
%t A206284 Complement[Range[200], v]      (* A206285 *)
%Y A206284 Cf. A206285, A206296.
%K A206284 nonn,new
%O A206284 1,1
%A A206284 Clark Kimberling (ck6(AT)evansville.edu), Feb 05 2012

%I A206076
%S A206076 1,1,1,1,5,1,3,1,7,5,3,1,9,3,5,1,17,7,9,5,21,3,13,1,15,9,7,3,
%T A206076 19,5,11,1,31,17,15,7,39,9,23,5,27,21,11,3,35,13,19,1,33,15,
%U A206076 17,9,41,7,25,3,29,19,13,5,37,11,21,1,65
%V A206076 1,-1,1,1,5,-1,3,-1,7,-5,3,1,9,-3,5,1,17,-7,9,5,21,-3,13,-1,15,-9,7,3,
%W A206076 19,-5,11,-1,31,-17,15,7,39,-9,23,-5,27,-21,11,3,35,-13,19,1,33,-15,
%X A206076 17,9,41,-7,25,-3,29,-19,13,5,37,-11,21,1,65
%N A206076 Numerator of p(n,-1/2), where p(n,x) is the polynomial given by A205073.
%C A206076 The denominators are given by A053644.  (See the Mathematica section.)
%t A206076 t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
%t A206076 b[n_] := Reverse[Table[x^k, {k, 0, n}]]
%t A206076 p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
%t A206076 Table[p[n, x], {n, 1, 15}]
%t A206076 Table[p[n, x] /. x -> 1, {n, 1, 120}]   (* A000120 *)
%t A206076 Table[p[n, x] /. x -> 2, {n, 1, 120}]   (* A000027 *)
%t A206076 Table[p[n, x] /. x -> 3, {n, 1, 120}]   (* A005836 *)
%t A206076 Table[p[n, x] /. x -> 4, {n, 1, 120}]   (* A000695 *)
%t A206076 Table[p[n, x] /. x -> -1, {n, 1, 120}]  (* A065359 *)
%t A206076 Table[p[n, x] /. x -> -2, {n, 1, 120}]  (* A053985 *)
%t A206076 Numerator[Table[p[n, x] /. x -> 1/2, {n, 1, 120}] ]     (* A030101 *)
%t A206076 Numerator[Table[p[n, x] /. x -> -1/2, {n, 1, 120}] ]    (* A206076 *)
%t A206076 Denominator[Table[p[n, x] /. x -> -1/2, {n, 1, 120}] ] (* A053644 *)
%Y A206076 Cf. A205073.
%K A206076 sign,new
%O A206076 1,5
%A A206076 Clark Kimberling (ck6(AT)evansville.edu), Feb 04 2012

%I A205958
%S A205958 1,1,1,1,2,2,3,3,48,16,40,40,270,270,945,63,129024,129024,64512,64512,
%T A205958 2016000,96000,528000,528000,144342000,28868400,187644600,20849400,
%U A205958 1787836050,1787836050,59594535,59594535,3999321544458240,121191561953280,1030128276602880
%N A205958 a(0) = 1 and a(n) = A180000(n)*a(floor(n/2))^2 for n > 0.
%C A205958 lcm(1,2,..,n) = (n!*a(n)) / ((n/2)!*a(n/2))^2.
%C A205958 lcm(1,2,..,n)*a(n) is a divisor of n! and n!/(lcm(1,2,..,n)*a(n)) is a square.
%o A205958 (Sage)
%o A205958 def A205958(n) :
%o A205958     if n == 0 : return 1
%o A205958     return A180000(n)*A205958(n//2)^2
%Y A205958 Cf. A180000, A025527, A205959.
%K A205958 nonn,new
%O A205958 0,5
%A A205958 Peter Luschny (peter(AT)luschny.de), Feb 04 2012

%I A206327
%S A206327 256,1344,1344,10504,16892,10504,83088,282996,282996,83088,673124,
%T A206327 4796092,10258408,4796092,673124,5432884,81302880,376249940,376249940,
%U A206327 81302880,5432884,43848120,1377662868,13817572060,30018231756
%N A206327 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206327 Table starts
%C A206327 .......256.........1344...........10504...............83088
%C A206327 ......1344........16892..........282996.............4796092
%C A206327 .....10504.......282996........10258408...........376249940
%C A206327 .....83088......4796092.......376249940.........30018231756
%C A206327 ....673124.....81302880.....13817572060.......2402295361876
%C A206327 ...5432884...1377662868....507368863264.....192331317667804
%C A206327 ..43848120..23340152044..18627016493468...15397943280509136
%C A206327 .353716772.395403752128.683798061454964.1232679968976699692
%H A206327 R. H. Hardin, <a href="/A206327/b206327.txt">Table of n, a(n) for n = 1..144</a>
%e A206327 Some solutions for n=4 k=3
%e A206327 ..2..0..0..2....2..1..3..0....0..2..0..3....2..0..1..0....1..2..3..3
%e A206327 ..3..2..1..0....0..2..1..2....3..0..3..0....3..1..2..1....1..3..1..0
%e A206327 ..0..3..2..1....1..3..2..1....0..2..1..3....0..3..0..2....3..2..0..3
%e A206327 ..3..0..3..2....0..1..0..2....0..3..2..0....3..2..3..2....2..1..3..2
%e A206327 ..1..1..0..3....2..3..2..3....1..0..3..1....2..0..1..3....0..2..0..3
%K A206327 nonn,tabl,new
%O A206327 1,1
%A A206327 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206326
%S A206326 43848120,23340152044,18627016493468,15397943280509136,
%T A206326 12867047450502167660,10794194502778261468304,
%U A206326 9069132565961324534394832,7624474056591556590163290412,6411545009003861373134720468216
%N A206326 Number of (n+1)X8 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206326 Column 7 of A206327
%H A206326 R. H. Hardin, <a href="/A206326/b206326.txt">Table of n, a(n) for n = 1..78</a>
%e A206326 Some solutions for n=4
%e A206326 ..0..2..0..0..3..3..0..3....3..0..1..2..2..2..1..0....3..0..3..0..1..0..1..0
%e A206326 ..1..0..3..2..1..0..3..1....1..2..0..1..0..3..2..1....1..3..2..3..0..2..0..1
%e A206326 ..2..1..0..3..2..1..0..3....2..3..1..2..1..0..3..2....3..2..3..0..2..0..3..0
%e A206326 ..3..2..1..0..3..2..1..0....0..0..2..3..2..1..0..3....1..3..0..3..1..2..0..1
%e A206326 ..1..3..2..1..0..3..2..1....2..0..3..0..3..2..1..0....2..0..1..0..2..3..1..3
%K A206326 nonn,new
%O A206326 1,1
%A A206326 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206325
%S A206325 5432884,1377662868,507368863264,192331317667804,73493501930345884,
%T A206325 28153923918860656368,10794194502778261468304,
%U A206325 4139628382451870628262520,1587705875694898193873275576,608959968033447679700747202336
%N A206325 Number of (n+1)X7 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206325 Column 6 of A206327
%H A206325 R. H. Hardin, <a href="/A206325/b206325.txt">Table of n, a(n) for n = 1..210</a>
%e A206325 Some solutions for n=4
%e A206325 ..3..2..0..0..1..0..3....1..3..3..2..2..0..3....2..0..3..3..3..1..1
%e A206325 ..2..3..2..1..2..1..0....3..2..1..0..3..1..0....0..3..2..1..0..3..2
%e A206325 ..1..2..1..0..1..0..1....2..1..3..1..0..2..1....1..0..3..2..1..0..3
%e A206325 ..3..1..2..1..2..1..0....0..2..0..2..1..3..2....2..1..0..3..2..1..0
%e A206325 ..0..2..0..2..0..2..1....2..0..1..3..2..0..3....3..2..1..0..3..2..1
%K A206325 nonn,new
%O A206325 1,1
%A A206325 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206324
%S A206324 673124,81302880,13817572060,2402295361876,419904864999804,
%T A206324 73493501930345884,12867047450502167660,2252845520588162061604,
%U A206324 394440987179669083025340,69060121294917710202087484
%N A206324 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206324 Column 5 of A206327
%H A206324 R. H. Hardin, <a href="/A206324/b206324.txt">Table of n, a(n) for n = 1..210</a>
%e A206324 Some solutions for n=4
%e A206324 ..0..0..2..1..3..0....2..0..2..3..2..2....3..0..2..3..2..1....2..2..1..2..1..1
%e A206324 ..2..1..3..2..1..0....1..2..1..2..1..3....0..3..0..1..3..2....0..3..2..1..3..2
%e A206324 ..1..2..0..3..2..1....1..0..3..0..2..1....1..0..2..3..0..3....1..0..3..2..0..3
%e A206324 ..3..1..2..1..3..2....2..1..0..3..0..2....2..1..3..0..3..2....2..1..0..3..1..0
%e A206324 ..0..2..3..2..1..3....1..2..1..0..2..3....0..2..0..3..1..3....0..2..1..0..2..1
%K A206324 nonn,new
%O A206324 1,1
%A A206324 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206323
%S A206323 83088,4796092,376249940,30018231756,2402295361876,192331317667804,
%T A206323 15397943280509136,1232679968976699692,98678331556478176912,
%U A206323 7899225044352464695308,632328349869410090441096
%N A206323 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206323 Column 4 of A206327
%H A206323 R. H. Hardin, <a href="/A206323/b206323.txt">Table of n, a(n) for n = 1..210</a>
%F A206323 Empirical: a(n) = 183*a(n-1) -12451*a(n-2) +431242*a(n-3) -8886139*a(n-4) +120866145*a(n-5) -1179014244*a(n-6) +8768675808*a(n-7) -51454745441*a(n-8) +240262258906*a(n-9) -885520958374*a(n-10) +2530542681588*a(n-11) -5428375302886*a(n-12) +8084083929724*a(n-13) -5930392323084*a(n-14) -7634354517794*a(n-15) +38890465996140*a(n-16) -95268108590731*a(n-17) +186624188350509*a(n-18) -301961470324629*a(n-19) +353763294618256*a(n-20) -167129467275573*a(n-21) -391779195765952*a(n-22) +1227500060831118*a(n-23) -1992750968159206*a(n-24) +2343202465599204*a(n-25) -2234194888988833*a(n-26) +1886349651989041*a(n-27) -1514677707006655*a(n-28) +1188167691894956*a(n-29) -881928373815179*a(n-30) +580988595478138*a(n-31) -317613769463907*a(n-32) +133650873588708*a(n-33) -37824298812889*a(n-34) +3800752301117*a(n-35) +2257980844041*a(n-36) -1251463484089*a(n-37) +259550198799*a(n-38) +3129578619*a(n-39) -14265223894*a(n-40) +2839493915*a(n-41) +9653750*a(n-42) -72347280*a(n-43) +5523888*a(n-44) +278636*a(n-45) -43200*a(n-46) for n>49
%e A206323 Some solutions for n=4
%e A206323 ..1..1..2..2..1....1..3..2..3..2....0..1..0..3..0....3..0..1..0..3
%e A206323 ..0..2..1..0..2....2..1..3..1..3....3..0..3..1..3....2..3..0..3..0
%e A206323 ..1..0..2..1..3....3..2..0..2..0....3..2..0..2..1....2..0..3..1..0
%e A206323 ..2..1..3..2..0....0..3..1..3..1....1..3..1..0..2....1..2..0..2..1
%e A206323 ..1..0..0..3..2....2..0..3..2..3....3..0..3..1..0....1..3..1..3..2
%K A206323 nonn,new
%O A206323 1,1
%A A206323 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206322
%S A206322 10504,282996,10258408,376249940,13817572060,507368863264,
%T A206322 18627016493468,683798061454964,25101386072770648,921428112379079596,
%U A206322 33823823595699973196,1241603426950143338776,45576678618716075262252
%N A206322 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206322 Column 3 of A206327
%H A206322 R. H. Hardin, <a href="/A206322/b206322.txt">Table of n, a(n) for n = 1..210</a>
%F A206322 Empirical: a(n) = 69*a(n-1) -1559*a(n-2) +15847*a(n-3) -85177*a(n-4) +268167*a(n-5) -545306*a(n-6) +718090*a(n-7) -456168*a(n-8) -215639*a(n-9) +632189*a(n-10) -402870*a(n-11) +99964*a(n-12) -248171*a(n-13) +364298*a(n-14) -135801*a(n-15) -12132*a(n-16) -208*a(n-17) +3772*a(n-18) +516*a(n-19) +120*a(n-20) for n>23
%e A206322 Some solutions for n=4
%e A206322 ..3..3..3..0....3..2..1..3....1..2..3..3....2..0..0..2....1..0..0..1
%e A206322 ..2..1..0..2....0..3..2..0....1..3..1..0....3..2..1..0....3..2..1..2
%e A206322 ..0..2..1..0....2..1..0..3....3..2..0..3....0..3..2..1....0..3..2..3
%e A206322 ..3..0..2..1....3..2..1..0....2..1..3..2....3..0..3..2....1..0..3..0
%e A206322 ..3..1..3..2....2..1..3..1....0..2..0..3....1..1..0..3....2..1..0..1
%K A206322 nonn,new
%O A206322 1,1
%A A206322 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206321
%S A206321 1344,16892,282996,4796092,81302880,1377662868,23340152044,
%T A206321 395403752128,6698395782004,113474612575628,1922321318016168,
%U A206321 32565148864444732,551670927155642340,9345598309321513584,158319393789428735572
%N A206321 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206321 Column 2 of A206327
%H A206321 R. H. Hardin, <a href="/A206321/b206321.txt">Table of n, a(n) for n = 1..210</a>
%F A206321 Empirical: a(n) = 24*a(n-1) -131*a(n-2) +200*a(n-3) -116*a(n-4) +35*a(n-5) -13*a(n-6) +2*a(n-7) for n>10
%e A206321 Some solutions for n=4
%e A206321 ..3..0..3....3..2..3....0..3..2....3..0..0....2..2..1....2..3..3....1..0..0
%e A206321 ..3..1..0....1..0..1....1..0..3....2..1..2....1..0..2....0..1..0....0..2..1
%e A206321 ..3..2..1....0..3..0....2..1..0....3..0..3....2..1..3....3..0..3....1..0..2
%e A206321 ..1..0..2....1..0..3....1..0..3....2..1..2....0..2..0....2..2..0....3..1..3
%e A206321 ..2..1..3....3..1..0....2..1..0....3..0..3....0..3..1....0..2..1....3..2..1
%K A206321 nonn,new
%O A206321 1,1
%A A206321 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206320
%S A206320 256,1344,10504,83088,673124,5432884,43848120,353716772,2853265228,
%T A206320 23015359308,185649551836,1497511097624,12079428217088,97436737226308,
%U A206320 785957567591456,6339798686936496,51138953368767152,412504036786299552
%N A206320 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206320 Column 1 of A206327
%H A206320 R. H. Hardin, <a href="/A206320/b206320.txt">Table of n, a(n) for n = 1..210</a>
%F A206320 Empirical: a(n) = 9*a(n-1) -5*a(n-2) -26*a(n-3) +42*a(n-4) +19*a(n-5) +42*a(n-6) +14*a(n-7) for n>8
%e A206320 Some solutions for n=4
%e A206320 ..0..0....3..2....1..1....1..1....3..2....3..2....1..3....0..3....2..2....3..1
%e A206320 ..1..1....1..1....2..2....3..3....0..1....0..3....2..2....0..0....0..3....0..1
%e A206320 ..3..2....3..2....1..0....1..0....3..2....3..0....0..3....3..2....3..2....3..2
%e A206320 ..0..3....0..3....0..3....2..1....1..1....1..3....2..1....2..1....0..3....0..0
%e A206320 ..1..0....1..0....1..0....1..3....0..2....2..1....1..0....3..2....1..0....3..1
%K A206320 nonn,new
%O A206320 1,1
%A A206320 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206319
%S A206319 256,16892,10258408,30018231756,419904864999804,28153923918860656368,
%T A206319 9069132565961324534394832,14057869699230976600019746011216
%N A206319 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly three clockwise edge increases
%C A206319 Diagonal of A206327
%e A206319 Some solutions for n=4
%e A206319 ..0..3..1..2..3....1..0..0..0..1....3..2..2..3..1....3..1..1..2..0
%e A206319 ..1..0..2..0..1....0..3..2..1..2....1..0..3..0..2....1..0..2..0..1
%e A206319 ..3..1..3..2..3....2..1..3..2..0....2..1..0..1..3....3..1..3..1..2
%e A206319 ..3..2..0..3..1....1..3..2..3..1....1..3..1..0..1....0..2..0..2..0
%e A206319 ..1..3..1..0..2....2..0..3..1..0....2..0..2..1..2....3..3..1..3..1
%K A206319 nonn,new
%O A206319 1,1
%A A206319 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206318
%S A206318 112,873,873,6831,14990,6831,53463,258452,258452,53463,418468,4466450,
%T A206318 9841886,4466450,418468,3275406,77132181,375274347,375274347,77132181,
%U A206318 3275406,25637095,1332632711,14309749540,31585648103,14309749540
%N A206318 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206318 Table starts
%C A206318 .......112..........873............6831...............53463
%C A206318 .......873........14990..........258452.............4466450
%C A206318 ......6831.......258452.........9841886...........375274347
%C A206318 .....53463......4466450.......375274347.........31585648103
%C A206318 ....418468.....77132181.....14309749540.......2656145209648
%C A206318 ...3275406...1332632711....545678293613.....223458785606871
%C A206318 ..25637095..23019152735..20808377921080...18792437854613234
%C A206318 .200665376.397663246671.793485464714845.1580709874695673480
%H A206318 R. H. Hardin, <a href="/A206318/b206318.txt">Table of n, a(n) for n = 1..144</a>
%e A206318 Some solutions for n=4 k=3
%e A206318 ..2..0..2..3....0..1..0..1....0..1..3..3....3..0..1..1....0..1..2..3
%e A206318 ..0..1..0..1....1..0..3..3....1..3..1..0....1..1..3..2....1..3..1..2
%e A206318 ..2..2..1..3....3..2..1..0....3..2..0..1....3..2..0..3....3..0..2..3
%e A206318 ..0..3..2..0....0..3..2..1....2..0..3..3....2..3..1..0....1..1..0..2
%e A206318 ..1..0..3..1....1..0..3..2....1..2..1..0....1..1..0..1....0..2..1..3
%K A206318 nonn,tabl,new
%O A206318 1,1
%A A206318 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206317
%S A206317 25637095,23019152735,20808377921080,18792437854613234,
%T A206317 16933907482183625564,15245666610426961469352,
%U A206317 13716254602125340475868528,12336975902231208201294106142,11094395430478243136970514760806
%N A206317 Number of (n+1)X8 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206317 Column 7 of A206318
%H A206317 R. H. Hardin, <a href="/A206317/b206317.txt">Table of n, a(n) for n = 1..210</a>
%e A206317 Some solutions for n=3
%e A206317 ..2..0..2..1..1..1..2..3....3..0..2..2..1..1..0..2....2..0..2..0..3..0..3..0
%e A206317 ..0..2..1..0..3..2..1..1....1..2..1..0..3..2..1..3....2..1..3..1..0..1..0..1
%e A206317 ..2..1..3..2..1..0..3..2....3..0..2..1..0..3..2..0....3..2..0..3..1..2..1..2
%e A206317 ..3..2..0..3..2..1..0..3....2..2..3..2..1..0..3..1....2..1..1..0..2..0..2..1
%K A206317 nonn,new
%O A206317 1,1
%A A206317 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206316
%S A206316 3275406,1332632711,545678293613,223458785606871,91342748577948857,
%T A206316 37329289780655758477,15245666610426961469352,
%U A206316 6226771909834248684544837,2542604116386182784400024312,1038308180737971719036992527154
%N A206316 Number of (n+1)X7 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206316 Column 6 of A206318
%H A206316 R. H. Hardin, <a href="/A206316/b206316.txt">Table of n, a(n) for n = 1..210</a>
%e A206316 Some solutions for n=4
%e A206316 ..3..0..1..1..1..2..3....3..2..1..3..2..1..2....1..3..3..3..1..0..1
%e A206316 ..3..2..0..3..2..3..2....0..3..2..0..3..2..0....3..2..1..0..2..1..0
%e A206316 ..2..3..1..0..3..0..3....2..0..3..1..0..3..1....1..3..2..1..0..2..1
%e A206316 ..3..0..3..1..0..2..0....1..3..1..2..1..0..2....3..0..3..2..1..0..2
%e A206316 ..1..2..1..2..1..3..2....1..0..2..0..2..1..3....0..1..0..3..2..1..0
%K A206316 nonn,new
%O A206316 1,1
%A A206316 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206315
%S A206315 418468,77132181,14309749540,2656145209648,492612276280514,
%T A206315 91342748577948857,16933907482183625564,3139181759716837800994,
%U A206315 581918239365125184573987,107870252944675279760828811
%N A206315 Number of (n+1)X6 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206315 Column 5 of A206318
%H A206315 R. H. Hardin, <a href="/A206315/b206315.txt">Table of n, a(n) for n = 1..210</a>
%e A206315 Some solutions for n=4
%e A206315 ..2..0..3..0..1..0....3..2..3..1..2..2....3..2..1..1..0..1....1..3..0..2..3..0
%e A206315 ..1..1..0..1..0..3....2..1..2..1..0..3....1..0..3..2..1..0....3..1..3..0..2..2
%e A206315 ..0..3..1..3..2..0....0..2..3..2..1..0....2..1..0..3..2..1....1..3..0..1..0..3
%e A206315 ..2..0..2..1..3..1....3..2..0..3..2..1....3..2..1..0..3..2....2..0..1..3..2..1
%e A206315 ..3..1..3..2..1..2....2..3..1..0..3..2....1..3..2..1..0..3....1..2..1..0..3..2
%K A206315 nonn,new
%O A206315 1,1
%A A206315 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206314
%S A206314 53463,4466450,375274347,31585648103,2656145209648,223458785606871,
%T A206314 18792437854613234,1580709874695673480,132941278551460873763,
%U A206314 11181607230811497171290,940424480197031998954820
%N A206314 Number of (n+1)X5 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206314 Column 4 of A206318
%H A206314 R. H. Hardin, <a href="/A206314/b206314.txt">Table of n, a(n) for n = 1..210</a>
%e A206314 Some solutions for n=4
%e A206314 ..3..2..2..1..1....0..1..2..2..1....1..1..0..3..2....1..2..2..1..0
%e A206314 ..2..1..0..3..2....1..0..1..3..2....3..2..1..0..3....1..0..3..2..1
%e A206314 ..3..2..1..0..3....0..1..0..1..3....1..3..2..1..0....2..1..0..3..2
%e A206314 ..2..3..2..1..0....1..2..1..2..0....3..2..3..2..1....0..3..2..0..3
%e A206314 ..2..0..3..2..1....2..3..2..1..2....2..0..1..3..2....1..0..3..1..0
%K A206314 nonn,new
%O A206314 1,1
%A A206314 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206313
%S A206313 6831,258452,9841886,375274347,14309749540,545678293613,
%T A206313 20808377921080,793485464714845,30257994130397426,1153827383666615864,
%U A206313 43998894149887669205,1677809124741057851874,63979878218662738329204
%N A206313 Number of (n+1)X4 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206313 Column 3 of A206318
%H A206313 R. H. Hardin, <a href="/A206313/b206313.txt">Table of n, a(n) for n = 1..210</a>
%F A206313 Empirical: a(n) = 31*a(n-1) +658*a(n-2) -14525*a(n-3) -51266*a(n-4) +1742315*a(n-5) +2568*a(n-6) -111724072*a(n-7) +306214837*a(n-8) +3458948497*a(n-9) -19921518115*a(n-10) -14103311618*a(n-11) +342323431454*a(n-12) -661578578970*a(n-13) -1100410213420*a(n-14) +4586308859937*a(n-15) -3284892664330*a(n-16) +12088917508494*a(n-17) -66798656945754*a(n-18) +4863204454404*a(n-19) +668152883487933*a(n-20) -1572909544825801*a(n-21) -104343036353394*a(n-22) +5200756598567925*a(n-23) -5763757360162957*a(n-24) -4460356071410713*a(n-25) +12820640624723848*a(n-26) +10571017688922653*a(n-27) -62236449619191771*a(n-28) +9498904108080154*a(n-29) +176626478662442442*a(n-30) -62348749427938505*a(n-31) -376071258209767792*a(n-32) +8405030630945943*a(n-33) +1182210730211193804*a(n-34) -543383167943355519*a(n-35) -2748832273517235471*a(n-36) +2511253093958231326*a(n-37) +4558534101790015100*a(n-38) -5030513212436441025*a(n-39) -5967019215596460086*a(n-40) +6526107983007343980*a(n-41) +5811203811965935737*a(n-42) -6638734846297707090*a(n-43) -4483028869982072147*a(n-44) +5416632074429772361*a(n-45) +3105155621945654476*a(n-46) -2358211188573072666*a(n-47) -1536238484904416722*a(n-48) +179738288581738082*a(n-49) +360896193919187643*a(n-50) -364600824797434408*a(n-51) -753172801379183144*a(n-52) +434773921978787104*a(n-53) +1374936398210390272*a(n-54) +649802974647708660*a(n-55) -419741593913265856*a(n-56) -751500276618548197*a(n-57) -383939729005838948*a(n-58) +23304274099593335*a(n-59) +198890112685590734*a(n-60) +144463750783801212*a(n-61) +31032683905835412*a(n-62) -37913693638469596*a(n-63) -51623968451360997*a(n-64) -34213660562008801*a(n-65) -11484588516621641*a(n-66) +1143116456421789*a(n-67) +3888906593680780*a(n-68) +2599794891913385*a(n-69) +1047824703702024*a(n-70) +153886579654709*a(n-71) -154662587901795*a(n-72) -161459478479969*a(n-73) -92779080676249*a(n-74) -39172361747793*a(n-75) -12089213690065*a(n-76) -2514036234254*a(n-77) -230505256750*a(n-78) +57500795847*a(n-79) +34151193540*a(n-80) +7629681820*a(n-81) -495077972*a(n-82) -904267877*a(n-83) -280667876*a(n-84) -73170285*a(n-85) -23446143*a(n-86) -5759543*a(n-87) -620246*a(n-88) -83104*a(n-89) -9272*a(n-90) +186*a(n-91) for n>92
%e A206313 Some solutions for n=4
%e A206313 ..2..1..1..0....2..1..1..0....2..1..3..3....2..0..2..3....0..1..2..3
%e A206313 ..0..3..2..1....1..3..2..1....1..2..1..0....0..1..0..1....1..3..1..2
%e A206313 ..1..0..3..2....0..1..3..2....1..3..2..1....2..2..1..3....3..0..2..3
%e A206313 ..0..2..0..3....1..2..1..3....1..0..3..2....0..3..2..0....1..1..0..2
%e A206313 ..1..3..1..0....3..3..2..0....0..1..0..3....1..0..3..1....0..2..1..3
%K A206313 nonn,new
%O A206313 1,1
%A A206313 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206312
%S A206312 873,14990,258452,4466450,77132181,1332632711,23019152735,
%T A206312 397663246671,6869398174303,118667836512792,2049943888882611,
%U A206312 35412251178185627,611735705434058145,10567560415916348461,182551476583398721925
%N A206312 Number of (n+1)X3 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206312 Column 2 of A206318
%H A206312 R. H. Hardin, <a href="/A206312/b206312.txt">Table of n, a(n) for n = 1..210</a>
%F A206312 Empirical: a(n) = 7*a(n-1) +178*a(n-2) +165*a(n-3) -2824*a(n-4) -4379*a(n-5) +21326*a(n-6) +27124*a(n-7) -88175*a(n-8) -686*a(n-9) +173432*a(n-10) -238738*a(n-11) -152649*a(n-12) +854730*a(n-13) -1223471*a(n-14) +1084153*a(n-15) -644627*a(n-16) +130265*a(n-17) +200912*a(n-18) -253639*a(n-19) +210358*a(n-20) -155058*a(n-21) +94508*a(n-22) -47283*a(n-23) +17559*a(n-24) -3954*a(n-25) -565*a(n-26) +581*a(n-27) -71*a(n-28) +6*a(n-29)
%e A206312 Some solutions for n=4
%e A206312 ..3..1..3....1..0..1....1..3..2....3..3..0....2..3..2....1..0..1....1..3..1
%e A206312 ..0..2..0....3..1..2....0..1..0....1..0..2....1..1..0....2..1..2....3..1..0
%e A206312 ..1..3..2....3..2..1....3..3..1....2..1..0....3..2..1....0..2..1....3..2..1
%e A206312 ..3..1..3....0..3..2....1..0..2....3..2..1....1..0..2....1..3..2....2..1..3
%e A206312 ..2..2..1....1..0..3....2..1..0....0..3..2....3..2..3....1..0..3....3..2..0
%K A206312 nonn,new
%O A206312 1,1
%A A206312 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206311
%S A206311 112,873,6831,53463,418468,3275406,25637095,200665376,1570637751,
%T A206311 12293615779,96223962193,753159288558,5895089964664,46141747460483,
%U A206311 361158332020412,2826840073596765,22126098426129671,173184268942327111
%N A206311 Number of (n+1)X2 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206311 Column 1 of A206318
%H A206311 R. H. Hardin, <a href="/A206311/b206311.txt">Table of n, a(n) for n = 1..210</a>
%F A206311 Empirical: a(n) = 5*a(n-1) +21*a(n-2) +13*a(n-3) -31*a(n-4) -17*a(n-5) +34*a(n-6) +2*a(n-7) -9*a(n-8) -a(n-9)
%e A206311 Some solutions for n=4
%e A206311 ..2..2....3..0....0..1....3..2....1..1....0..1....3..1....2..0....1..0....2..2
%e A206311 ..0..3....2..2....1..3....0..3....3..2....1..2....3..2....1..1....0..1....0..3
%e A206311 ..3..0....0..3....2..1....3..0....0..3....0..2....1..3....3..2....1..2....2..1
%e A206311 ..0..1....1..0....1..0....1..3....2..0....2..1....3..0....2..1....2..1....1..0
%e A206311 ..2..3....2..1....3..1....2..1....0..1....0..3....1..1....3..2....0..2....2..1
%K A206311 nonn,new
%O A206311 1,1
%A A206311 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206310
%S A206310 112,14990,9841886,31585648103,492612276280514,37329289780655758477,
%T A206310 13716254602125340475868528,24435414110066461941469656297723
%N A206310 Number of (n+1)X(n+1) 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two, and every 2X2 determinant nonzero
%C A206310 Diagonal of A206318
%e A206310 Some solutions for n=4
%e A206310 ..1..1..0..2..1....0..3..2..1..1....1..2..2..1..0....3..3..2..1..1
%e A206310 ..3..2..1..3..2....2..1..0..3..2....1..0..3..2..1....2..1..0..3..2
%e A206310 ..2..3..2..0..3....3..2..1..0..3....2..1..0..3..2....3..2..1..0..3
%e A206310 ..0..1..0..2..0....1..3..2..1..0....0..3..2..0..3....0..3..2..1..0
%e A206310 ..1..3..2..0..1....1..0..3..2..1....1..0..3..1..0....2..0..3..2..1
%K A206310 nonn,new
%O A206310 1,1
%A A206310 R. H. Hardin (rhhardin(AT)att.net) Feb 06 2012

%I A206301
%S A206301 1,1,2,4,9,19,43,93,207,453,1003,2200,4860,10681,23552,51819,114186,
%T A206301 251326,553634,1218857,2684461,5910729,13016952,28662693,63120135,
%U A206301 138991543,306076520,673995311,1484205869,3268315926,7197126602,15848588048,34899932674
%N A206301  G.f. satisfies: A(x) = Sum_{n>=0} x^n * Product_{k=1..n} A(x^k).
%F A206301  G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1+x*A(x) - x*A(x^2)/(1+x*A(x^2) - x*A(x^3)/(1+x*A(x^3) -...)))), a recursive continued fraction.
%e A206301  G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 43*x^6 + 93*x^7 +...
%e A206301 such that, by definition,
%e A206301 A(x) = 1 + x*A(x) + x^2*A(x)*A(x^2) + x^3*A(x)*A(x^2)*A(x^3) + x^4*A(x)*A(x^2)*A(x^3)*A(x^4) + x^5*A(x)*A(x^2)*A(x^3)*A(x^4)*A(x^5) +...
%e A206301 The coefficients in Product_{k=1..n} A(x^k) begin:
%e A206301 n=2: [1, 1, 3, 5, 13, 25, 60, 124, 285, 609, 1369, 2970, 6611, ...];
%e A206301 n=3: [1, 1, 3, 6, 14, 28, 67, 139, 316, 683, 1523, 3317, 7369, ...];
%e A206301 n=4: [1, 1, 3, 6, 15, 29, 70, 145, 332, 713, 1596, 3468, 7717, ...];
%e A206301 n=5: [1, 1, 3, 6, 15, 30, 71, 148, 338, 728, 1627, 3540, 7868, ...];
%e A206301 n=6: [1, 1, 3, 6, 15, 30, 72, 149, 341, 734, 1642, 3570, 7941, ...];
%e A206301 n=7: [1, 1, 3, 6, 15, 30, 72, 150, 342, 737, 1648, 3585, 7971, ...]; ...
%o A206301  (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*prod(k=1, m, subst(A, x, x^k +x*O(x^n))))); polcoeff(A, n)}
%o A206301 for(n=0, 35, print1(a(n), ", "))
%Y A206301  Cf. A206302, A091865.
%K A206301 nonn,new
%O A206301 0,3
%A A206301 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2012

%I A206246
%S A206246 1,3,7,13,21,31,43,91,111,183,211,241,273,381,421,553,601,651,703,
%T A206246 1261,1333,1561,1641,2863,2971,3081,3193,4291,4423,5403,5551,6973,
%U A206246 7141,8011,8191,8743,8931,11991,12211,13341,13573,14281,14521,15253,15501,15751,16003
%N A206246 Numbers n such that the greatest prime divisor p of n^2+1 has the property that (p - n)^2 + 1 = p.
%C A206246 For the n > 1 in this sequence, n^2+1 is composite. The corresponding primes p are A002496(n) repeated two times for n > 1 : {2, 5, 5, 17, 17, 37, 37, 101, 101, 197,...}.
%C A206246 Because this sequence is connected with A002496, it is conjectured that the set of this numbers is infinite.
%e A206246 31 is in the sequence because 31^2 + 1 = 2*13*37 and (37 - 31)^2 + 1 = 37.
%e A206246 43 is in the sequence because 43^2 + 1 = 2*5*5*37 and (37 - 43)^2 + 1 = 37.
%p A206246 with(numtheory):for n from 1 to 20000 do:x:=n^2+1:y:=factorset(x):n1:=nops(y):p:=y[n1]:q:=(p-n)^2+1:if q=p then printf(`%d, `,n): else fi:od:
%Y A206246 Cf. A002496, A134406.
%K A206246 nonn,new
%O A206246 1,2
%A A206246 Michel Lagneau (mn.lagneau2(AT)orange.fr), Feb 05 2012

%I A185139
%S A185139 1,3,10,7,25,91,15,56,210,792,31,119,456,1749,6721,63,246,957,3718,
%T A185139 14443,56134,127,501,1969,7722,30251,118456,463828,255,1012,4004,
%U A185139 15808,62322,245480,966416,3803648,511,2035,8086,32071,127024,502588,1987096,7852453,31020445,1023,4082,16263
%N A185139 Triangle T(n,k) = sum_{i=1..n} 2^(i-1)*C(n+2*k-i-1, k-1), 1 <= k <= n.
%C A185139 The first term of the m-th row is 2^m-1.
%H A185139 V. Shevelev and P. Moses, <a href="http://arxiv.org/abs/1112.5715">On a sequence of polynomials with hypothetically integer coefficients</a>
%F A185139 2*T_n(k) = T_(n-1)(k+1) + C(n+2*k-1,k).
%F A185139 T_n(k) = T_(n-2)(k+1) + C(n+2*k-1,k).
%F A185139 T_n(k) = 2*T_(n-1)(k) + C(n+2*k-2,k-1).
%F A185139 T_n(k+1) = 4*T_n(k) - (n/k)*C(n+2*k-1,k-1).
%e A185139 Triangle begins
%e A185139 1,
%e A185139 3,     10,
%e A185139 7,     25,    91,
%e A185139 15,    56,    210,  792,
%e A185139 31,    119,   456,  1749,  6721,
%e A185139 63,    246,   957,  3718,  14443,  56134,
%e A185139 127,   501,   1969, 7722,  30251,  118456, 463828,
%e A185139 255,   1012,  4004, 15808, 62322,  245480, 966416,  3803648,
%e A185139 511,   2035,  8086, 32071, 127024, 502588, 1987096, 7852453, 31020445,
%e A185139 ...
%Y A185139 Cf. A174531.
%K A185139 nonn,tabl,new
%O A185139 1,2
%A A185139 Vladimir Shevelev and Peter Moses, Feb 04 2012

%I A206294
%S A206294 1,0,1,0,3,1,0,6,6,1,0,10,21,9,1,0,15,56,45,12,1,0,21,126,165,78,15,1,
%T A206294 0,28,252,495,364,120,18,1,0,36,462,1287,1365,680,171,21,1,0,45,792,
%U A206294 3003,4368,3060,1140,231,24,1
%N A206294 Riordan array (1, x/(1-x)^3).
%C A206294 Triangle T(n,k), read by rows, given by (0, 3, -1, 2/3, -1/6, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A206294 Row sums are A052529(n).
%F A206294 T(n,0) = 0^n, T(n,k) = C(n-1+2k, n-k) for k>0.
%F A206294 T(n,n) = 1, T(k+1,k) = 3*k = A008585(k), T(k+2,k) = A081266(k).
%F A206294 Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A052529(n), A052910(n) for x = 0, 1, 2 respectively.
%F A206294 G.f.: (1-x)^3/((1-x)^3-y*x).
%e A206294 Triangle begins :
%e A206294 1
%e A206294 0, 1
%e A206294 0, 3, 1
%e A206294 0, 6, 6, 1
%e A206294 0, 10, 21, 9, 1
%e A206294 0, 15, 56, 45, 12, 1
%e A206294 0, 21, 126, 165, 78, 15, 1
%e A206294 0, 28, 252, 495, 364, 120, 18, 1
%e A206294 0, 36, 462, 1287, 1365, 680, 171, 21, 1
%e A206294 0, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1
%e A206294 0, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1
%e A206294 0, 66, 2002, 12870, 31824, 38760, 26324, 10626, 2600, 378, 30, 1
%o A206294  (PARI) {T(n,k)=polcoeff(1/(1-x+x*O(x^(n-k)))^(3*k),n-k)}
%o A206294 (PARI) {T(n,k)=polcoeff(polcoeff((1-x)^3/((1-x)^3-y*x +x*O(x^n)),n,x),k,y)}
%o A206294 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A206294 Cf. Columns : A000007, A000217 (triangular numbers), A000389, A000581, A001288, A010967..(+3)..A011000, A017714..(+3)..A017762.
%Y A206294 Cf. A127893.
%K A206294 easy,nonn,tabl,new
%O A206294 0,5
%A A206294 DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 05 2012

%I A206290
%S A206290 1,1,2,3,5,7,12,17,29,44,77,114,218,330,617,987,1913,2968,6068,9500,
%T A206290 19263,31399,64268,101702,218891,348559,735823,1205239,2576727,
%U A206290 4119884,9100854,14588992,31841260,52163378,114485092,183947681,414704366,667453931,1487920000
%N A206290 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).
%C A206290 Compare to the g.f. of partition numbers (A000041): Sum_{n>=0} Product_{k=1..n} x/(1 - x^k).
%F A206290 G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
%F A206290 (1) G_n(x) = Series_Reversion( x/(1 + x^n) ),
%F A206290 (2) G_n(x) = x + x*G_n(x)^n,
%F A206290 (3) G_n(x) = Sum_{k>=0} binomial(n*k+1, k) * x^(n*k+1) / (n*k+1).
%e A206290 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*x^7 +...
%e A206290 such that, by definition,
%e A206290 A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
%e A206290 where G_n( x/(1 + x^n) ) = x.
%e A206290 The first few expansions of G_n(x) begin:
%e A206290 G_1(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...
%e A206290 G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...
%e A206290 G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...
%e A206290 G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...
%e A206290 G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...
%e A206290 G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...
%e A206290 G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...
%e A206290 Note that G_n(x) = x + x*G_n(x)^n.
%o A206290 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x/(1+x^k+x*O(x^n))))),n)}
%o A206290 for(n=0,45,print1(a(n),", "))
%Y A206290 Cf. A206289, A194560.
%K A206290 nonn,new
%O A206290 0,3
%A A206290 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206293
%S A206293 1,1,2,5,18,78,415,2467,16212,114623,863229,6858780,57156213,
%T A206293 497147291,4497291265,42189445764,409478828567,4103901097024,
%U A206293 42403116824997,451059832858894,4933844398096693,55436157047213427,639215949145395559,7557505365363885063
%N A206293 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/A(x^k) ).
%F A206293 G.f.: A(x) = Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
%F A206293 (1) G_n(x) = Series_Reversion( x/A(x^n) ),
%F A206293 (2) G_n(x) = x * A( G_n(x)^n ).
%e A206293 G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 78*x^5 + 415*x^6 + 2467*x^7 +...
%e A206293 such that, by definition,
%e A206293 A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
%e A206293 where G_n(x) satisfies: G_n( x/A(x^n) ) = x.
%e A206293 The first few expansions of G_n(x) begin:
%e A206293 G_1(x) = x + x^2 + 3*x^3 + 12*x^4 + 59*x^5 + 329*x^6 + 2035*x^7 +...
%e A206293 G_2(x) = x + x^3 + 4*x^5 + 22*x^7 + 144*x^9 + 1045*x^11 + 8159*x^13 +...
%e A206293 G_3(x) = x + x^4 + 5*x^7 + 35*x^10 + 289*x^13 + 2626*x^16 +...
%e A206293 G_4(x) = x + x^5 + 6*x^9 + 51*x^13 + 510*x^17 + 5597*x^21 +...
%e A206293 G_5(x) = x + x^6 + 7*x^11 + 70*x^16 + 823*x^21 + 10608*x^26 +...
%e A206293 G_6(x) = x + x^7 + 8*x^13 + 92*x^19 + 1244*x^25 + 18434*x^31 +...
%e A206293 G_7(x) = x + x^8 + 9*x^15 + 117*x^22 + 1789*x^29 + 29975*x^36 +...
%e A206293 where G_n(x) = x*A( G_n(x)^n ).
%o A206293 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,prod(k=1,m,serreverse(x/subst(A,x,x^k +x*O(x^n))))));polcoeff(A,n)}
%o A206293 for(n=0,45,print1(a(n),", "))
%Y A206293 Cf. A206290.
%K A206293 nonn,new
%O A206293 0,3
%A A206293 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206289
%S A206289 1,1,2,4,10,25,73,214,679,2189,7331,24867,86269,302144,1072621,
%T A206289 3837768,13853674,50319789,183941789,675731105,2494370326,9244865453,
%U A206289 34394851701,128390336942,480749791772,1805161153783,6795744287172,25643914891284,96980809856731
%N A206289 G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x*(1 - x^k) ).
%C A206289 Compare to the g.f. of partitions of n into distinct parts (A000009): Sum_{n>=0} Product_{k=1..n} x*(1 + x^k).
%F A206289 G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:
%F A206289 (1) G_n(x) = Series_Reversion( x*(1 - x^n) ),
%F A206289 (2) G_n(x) = x + x*G_n(x)^(n+1),
%F A206289 (3) G_n(x) = Sum_{k>=0} binomial(n*k+k+1, k) * x^(n*k+1) / (n*k+k+1).
%e A206289 G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 25*x^5 + 73*x^6 + 214*x^7 +...
%e A206289 such that, by definition,
%e A206289 A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
%e A206289 where G_n( x*(1 - x^n) ) = x.
%e A206289 The first few expansions of G_n(x) begin:
%e A206289 G_1(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 +...+ A000108(n)*x^(n+1) +...
%e A206289 G_2(x) = x + x^3 + 3*x^5 + 12*x^7 + 55*x^9 +...+ A001764(n)*x^(2*n+1) +...
%e A206289 G_3(x) = x + x^4 + 4*x^7 + 22*x^10 + 140*x^13 +...+ A002293(n)*x^(3*n+1) +...
%e A206289 G_4(x) = x + x^5 + 5*x^9 + 35*x^13 + 285*x^17 +...+ A002294(n)*x^(4*n+1) +...
%e A206289 G_5(x) = x + x^6 + 6*x^11 + 51*x^16 + 506*x^21 +...+ A002295(n)*x^(5*n+1) +...
%e A206289 G_6(x) = x + x^7 + 7*x^13 + 70*x^19 + 819*x^25 +...+ A002296(n)*x^(6*n+1) +...
%e A206289 Note that G_n(x) = x + x*G_n(x)^(n+1).
%o A206289 (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x*(1-x^k+x*O(x^n))))),n)}
%o A206289 for(n=0,35,print1(a(n),", "))
%Y A206289 Cf. A206290, A194560.
%K A206289 nonn,new
%O A206289 0,3
%A A206289 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206298
%S A206298 1,2,0,2,1,2,4,2,2,6,4,4,10,6,8,16,9,10,24,14,16,36,20,
%T A206298 24,53,30,32,76,43,48,108,60,68,150,84,92,206,114,128,280,
%U A206298 155,172,376,208,228,504,276,304,668,366,400,878,480,524,1148
%V A206298 1,-2,0,2,-1,-2,4,-2,-2,6,-4,-4,10,-6,-8,16,-9,-10,24,-14,-16,36,-20,
%W A206298 -24,53,-30,-32,76,-43,-48,108,-60,-68,150,-84,-92,206,-114,-128,280,
%X A206298 -155,-172,376,-208,-228,504,-276,-304,668,-366,-400,878,-480,-524,1148
%N A206298 McKay-Thompson series of class 24C for the Monster group with a(0) = -2.
%C A206298 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A206298 M. Somos, <a href="http://cis.csuohio.edu/~somos/multiq.pdf">Introduction to Ramanujan theta functions</a>
%H A206298 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A206298 Expansion of (1/q) * (phi(-q) * psi(q^4)) / (phi(-q^3) * psi(q^12)) in powers of q where phi(), psi() are Ramanujan theta functions.
%F A206298 Expansion of eta(q)^2 * eta(q^6) * eta(q^8)^2 * eta(q^12) / (eta(q^2) * eta(q^3)^2 * eta(q^4) * eta(q^24)^2) in powers of q.
%F A206298 Euler transform of period 24 sequence [ -2, -1, 0, 0, -2, 0, -2, -2, 0, -1, -2, 0, -2, -1, 0, -2, -2, 0, -2, 0, 0, -1, -2, 0, ...].
%F A206298 G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3 / f(t) where q = exp(2 pi i t).
%F A206298 a(n) = A058573(n) unless n = 0.
%e A206298 1/q - 2 + 2*q^2 - q^3 - 2*q^4 + 4*q^5 - 2*q^6 - 2*q^7 + 6*q^8 - 4*q^9 + ...
%o A206298 (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A) * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A)^2), n))}
%Y A206298 Cf. A058573.
%K A206298 sign,new
%O A206298 -1,2
%A A206298 Michael Somos, Feb 05 2012

%I A206282
%S A206282 1,1,1,4,5,1,9,11,4,25,31,9,64,79,25,169,209,64,441,545,169,
%T A206282 1156,1429,441,3025,3739,1156,7921,9791,3025,20736,25631,7921,
%U A206282 54289,67105,20736,142129,175681,54289,372100,459941,142129,974169
%V A206282 1,1,-1,-4,-5,1,9,11,-4,-25,-31,9,64,79,-25,-169,-209,64,441,545,-169,
%W A206282 -1156,-1429,441,3025,3739,-1156,-7921,-9791,3025,20736,25631,-7921,
%X A206282 -54289,-67105,20736,142129,175681,-54289,-372100,-459941,142129,974169
%N A206282 a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.
%C A206282 This satisfies the same recurrence as Dana Scott's sequence A048736.
%H A206282 Reinhard Zumkeller, <a href="/A206282/b206282.txt">Table of n, a(n) for n = 1..5000</a>
%H A206282 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,2,0,0,1).
%F A206282 G.f.: x * (1 + x - x^2 - 2*x^3 - 3*x^4 - x^5 - x^6 - x^7) / (1 + 2*x^3 - 2*x^6 - x^9).
%F A206282 a(-5 - n) = a(n) = a(n+2) * a(n-2) - a(n+1) * a(n-1).
%F A206282 a(3*n) = (-1)^n * F(n)^2, a(3*n + 1) = (-1)^n * F(n + 2)^2 where F =
%F A206282 Fibonacci A000045.
%F A206282 a(6*n - 4) = - A110034(2*n), a(6*n - 1) = - A110035(2*n), a(3*n + 2) = (-1)^n * A126116(2*n + 3).
%e A206282 x + x^2 - x^3 - 4*x^4 - 5*x^5 + x^6 + 9*x^7 + 11*x^8 - 4*x^9 - 25*x^10 + ...
%o A206282 (PARI) {a(n) = local(k = n\3); (-1)^k * if( n%3 == 0, fibonacci( k )^2, if (n%3 ==1, fibonacci( k+2 )^2, fibonacci( k ) * fibonacci( k+3 ) + fibonacci( k+1 ) * fibonacci( k+2 )))}
%o A206282 (Haskell)
%o A206282 a206282 n = a206282_list !! (n-1)
%o A206282 a206282_list = 1 : 1 : -1 : -4 :
%o A206282    zipWith div
%o A206282      (zipWith (+)
%o A206282        (zipWith (*) (drop 3 a206282_list)
%o A206282                     (drop 1 a206282_list))
%o A206282        (drop 2 a206282_list))
%o A206282      a206282_list
%o A206282 -- Same program as in A048736, see comment.
%o A206282 -- Reinhard Zumkeller, Feb 08 2012
%Y A206282 Cf. A000045, A048736, A110034, A110035, A126116.
%K A206282 sign,easy,new
%O A206282 1,4
%A A206282 Michael Somos, Feb 05 2012

%I A198176
%S A198176 1,6,168,26880
%N A198176 Number of orthogonal cocycles in Z^2(Z^t_2,Z_2).
%D A198176 Horadam, K. J., Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp.  See p. 132.
%K A198176 nonn,new
%O A198176 1,2
%A A198176 N. J. A. Sloane (njas(AT)research.att.com), Feb 05 2012

%I A185120
%S A185120 2,71,8281828459045235360287471,
%T A185120 352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059
%N A185120 Cut decimal expansion of e (A001113) into pieces that are primes, each prime being greater in length than the last.
%C A185120 If we omit the condition that the terms increase in length, the sequence begins 2, 7, ?. Ignacio Larrosa Canestro reports that the next term has at least 169 digits.
%Y A185120 Cf. A001113, A073246, A104843.
%Y A185120 A subsequence of A198188. [From M. F. Hasler, Feb 05 2012]
%K A185120 nonn,base,new
%O A185120 1,1
%A A185120 N. J. A. Sloane (njas(AT)research.att.com), Feb 05 2012
%E A185120 a(4) from Ignacio Larrosa Canestro, Feb 05 2012

%I A206159
%S A206159 1,11,13,17,19,31,41,61,71,101,113,121,131,151,181,191,199,211,311,
%T A206159 313,331,661,811,881,911,919,991,1111,1117,1151,1171,1181,1511,1777,
%U A206159 1811,1999,2111,2221,3313,3331,4111,4441,6661,7177,7717,8111,9199,10111,11113
%N A206159 Numbers needing at most two digits to write all positive divisors in decimal representation.
%C A206159 A095048(a(n)) <= 2; apparently 1, 121 and 1111 are the only nonprime terms.
%H A206159 Reinhard Zumkeller, <a href="/A206159/b206159.txt">Table of n, a(n) for n = 1..476</a>
%o A206159 (Haskell)
%o A206159 a206159 n = a206159_list !! (n-1)
%o A206159 a206159_list = filter ((<= 2) . a095048) [1..]
%Y A206159 Cf. A027750, A031955, A011531, A106101, A004022, A062634.
%K A206159 nonn,base,new
%O A206159 1,2
%A A206159 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 05 2012

%I A206281
%S A206281 181,199,317,3529,3733,4177,4663,9049,15329,15991,19577,24907,43607,
%T A206281 47017,58073,84223,86843,146191,152417,156623,175543,217559,227671,
%U A206281 288461,308999,323077,331249,333323,354301,390289,397037,407249,474923,476137,491059,520339
%N A206281 Smallest of five consecutive primes whose sum is a square.
%e A206281 a(4) = 3529.  The next four primes are 3533, 3539, 3541,and 3547, and the sum of those five primes = 17689 = 133^2.
%t A206281 Transpose[Select[Partition[Prime[Range[80000]],5,1],IntegerQ[Sqrt[ Total[#]]]&]][[1]]
%Y A206281 Cf. A061275, A206279, A206280.
%K A206281 nonn,easy,new
%O A206281 1,1
%A A206281 Harvey P. Dale (hpd1(AT)nyu.edu), Feb 05 2012

%I A206280
%S A206280 5,73,137,433,569,1217,5171,15859,16631,32027,35677,37619,39191,45767,
%T A206280 59029,63997,65011,77813,92401,103669,186601,196201,230387,237161,
%U A206280 261089,273517,439559,463747,484397,488573,505511,514079,519803,538739,544627,633599
%N A206280 Smallest of four consecutive primes whose sum is a square.
%e A206280 a(4) = 433. The next three primes are 439, 443, and 449, and the sum of those four primes = 1764 = 42^2.
%t A206280 Transpose[Select[Partition[Prime[Range[80000]],4,1],IntegerQ[Sqrt[ Total[#]]]&]][[1]]
%Y A206280 Cf. A061275, A206279, A206281.
%K A206280 nonn,easy,new
%O A206280 1,1
%A A206280 Harvey P. Dale (hpd1(AT)nyu.edu), Feb 05 2012

%I A206279
%S A206279 13,37,277,313,613,7591,8209,12157,14557,15679,16267,23053,32233,
%T A206279 42953,44887,55213,81013,94687,105649,106397,135241,203317,221401,
%U A206279 225769,235747,245941,258707,287671,306541,331333,342049,346111,347443,393853,479191,488827
%N A206279 Smallest of three consecutive primes whose sum is a square.
%e A206279 a(4) = 313.  The next two primes are 317 and 331, and 313 + 317 + 331 = 961 = 31^2.
%t A206279 Transpose[Select[Partition[Prime[Range[50000]],3,1],IntegerQ[ Sqrt[ Total[#]]]&]][[1]]
%Y A206279 Cf. A061275, A206280, A206281.
%K A206279 nonn,easy,new
%O A206279 1,1
%A A206279 Harvey P. Dale (hpd1(AT)nyu.edu), Feb 05 2012

%I A206278
%S A206278 0,0,128,1024,6656,53248,387072,3096576,24092672,192741376,1530822656,
%T A206278 12246581248,97793998848,782351990784,6255953838080,50047630704640,
%U A206278 400335237545984,3202681900367872,25620722214764544,204965777718116352,1639714493699194880,13117715949593559040,104941539947077173248,839532319576617385984
%N A206278 Total number of triangles in Cayley graph Cay(Z_{2^n}, QR*(2^n)).
%D A206278 Giudici, Reinaldo E.;  and Olivieri, Aurora A.; Quadratic modulo 2n Cayley graphs. Discrete Math. 215 (2000), no. 1-3, 73-79. See T(n) in Theorem 3.1.
%p A206278 f:=n-> if n mod 2 = 1 then
%p A206278 (1/45)*(2^(3*(n-1))+5*2^(2*n-1)-7*2^(n+2));
%p A206278 else
%p A206278 (1/45)*(2^(3*(n-1))+5*2^(2*n)-7*2^(n+4));
%p A206278 fi;
%p A206278 [seq(f(n),n=3..40)];
%K A206278 nonn,new
%O A206278 3,3
%A A206278 N. J. A. Sloane (njas(AT)research.att.com), Feb 05 2012

%I A206277
%S A206277 81,306,306,1233,972,1233,4884,3168,3168,4884,19509,11205,12429,11205,
%T A206277 19509,77580,35922,39306,39306,35922,77580,309057,123357,133866,
%U A206277 145824,133866,123357,309057,1230480,408198,437340,495858,495858,437340,408198
%N A206277 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206277 Table starts
%C A206277 ......81.....306....1233.....4884.....19509.....77580.....309057.....1230480
%C A206277 .....306.....972....3168....11205.....35922....123357.....408198.....1376754
%C A206277 ....1233....3168...12429....39306....133866....437340....1528920.....4972464
%C A206277 ....4884...11205...39306...145824....495858...1824315....6254364....22950822
%C A206277 ...19509...35922..133866...495858...2243139...8405508...33297948...125846844
%C A206277 ...77580..123357..437340..1824315...8405508..39044748..167821686...751671621
%C A206277 ..309057..408198.1528920..6254364..33297948.167821686..964315869..4586324364
%C A206277 .1230480.1376754.4972464.22950822.125846844.751671621.4586324364.27362301828
%H A206277 R. H. Hardin, <a href="/A206277/b206277.txt">Table of n, a(n) for n = 1..262</a>
%e A206277 Some solutions for n=4 k=3
%e A206277 ..2..1..1..0....2..2..1..1....1..0..1..1....1..0..0..1....0..2..0..2
%e A206277 ..2..0..0..2....1..1..0..0....1..1..2..1....2..1..1..2....0..0..1..1
%e A206277 ..2..0..0..2....1..1..0..0....1..1..1..0....1..1..1..2....0..0..1..1
%e A206277 ..0..1..1..0....2..2..1..0....0..1..1..0....1..1..2..1....0..1..2..2
%e A206277 ..0..1..1..0....2..2..2..1....1..0..0..1....1..2..1..1....1..2..2..2
%K A206277 nonn,tabl,new
%O A206277 1,1
%A A206277 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206276
%S A206276 309057,408198,1528920,6254364,33297948,167821686,964315869,
%T A206276 4586324364,22656222912,107944182948,562553467470,2820402520110,
%U A206276 14881797312963,75177032639256,401169616195323,2040321270376866,10870614344272152
%N A206276 Number of (n+1)X8 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206276 Column 7 of A206277
%H A206276 R. H. Hardin, <a href="/A206276/b206276.txt">Table of n, a(n) for n = 1..210</a>
%e A206276 Some solutions for n=4
%e A206276 ..0..0..2..2..0..0..2..1....0..2..2..2..1..2..2..1....1..2..2..1..2..0..0..1
%e A206276 ..1..1..0..0..2..2..0..0....0..2..2..0..2..2..2..1....0..2..2..2..0..0..0..1
%e A206276 ..1..1..0..0..2..2..0..0....2..0..0..2..2..2..0..2....2..0..2..2..0..0..1..2
%e A206276 ..0..0..1..1..0..0..1..1....0..0..0..2..2..1..2..2....2..2..0..0..1..1..2..2
%e A206276 ..0..0..1..1..0..0..1..1....0..0..1..0..0..2..2..2....2..2..0..0..1..1..1..0
%K A206276 nonn,new
%O A206276 1,1
%A A206276 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206275
%S A206275 77580,123357,437340,1824315,8405508,39044748,167821686,751671621,
%T A206275 3186738756,14344492080,63040672158,293272589487,1300384478688,
%U A206275 6029134019499,26776062237576,124337331861300,551531622746376
%N A206275 Number of (n+1)X7 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206275 Column 6 of A206277
%H A206275 R. H. Hardin, <a href="/A206275/b206275.txt">Table of n, a(n) for n = 1..210</a>
%e A206275 Some solutions for n=4
%e A206275 ..2..2..2..1..1..1..1....0..0..2..0..0..2..2....1..2..0..1..2..2..2
%e A206275 ..0..1..1..2..1..1..0....2..1..0..0..0..2..2....1..2..1..2..2..2..0
%e A206275 ..0..1..1..1..2..2..1....1..0..0..0..1..0..0....1..0..2..2..2..1..2
%e A206275 ..1..2..1..1..2..2..1....1..0..0..2..0..0..0....2..1..2..2..0..2..2
%e A206275 ..1..1..0..0..1..1..2....2..1..1..0..0..0..2....0..1..1..0..2..2..2
%K A206275 nonn,new
%O A206275 1,1
%A A206275 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206274
%S A206274 19509,35922,133866,495858,2243139,8405508,33297948,125846844,
%T A206274 520171458,2032882188,8441030760,32995135068,137694091206,
%U A206274 538836368046,2242526518020,8781930040704,36621430354950,143540058923604
%N A206274 Number of (n+1)X6 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206274 Column 5 of A206277
%H A206274 R. H. Hardin, <a href="/A206274/b206274.txt">Table of n, a(n) for n = 1..210</a>
%e A206274 Some solutions for n=4
%e A206274 ..2..0..1..1..0..1....0..0..2..0..0..1....1..2..1..1..0..1....2..1..1..2..2..0
%e A206274 ..2..1..2..2..1..1....0..1..0..0..0..1....1..0..2..2..1..1....1..0..0..1..1..2
%e A206274 ..1..2..2..2..1..1....2..0..0..0..1..0....2..1..2..2..1..1....1..0..0..1..1..2
%e A206274 ..1..2..2..0..2..0....2..0..0..1..2..2....0..1..1..0..2..0....2..1..1..0..0..1
%e A206274 ..2..0..0..1..2..2....0..1..1..2..2..2....0..0..2..1..2..0....1..1..1..0..0..1
%K A206274 nonn,new
%O A206274 1,1
%A A206274 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206273
%S A206273 4884,11205,39306,145824,495858,1824315,6254364,22950822,80340090,
%T A206273 297576123,1049291592,3857539692,13593707712,50108727009,177035467728,
%U A206273 652290312486,2303858318196,8487080173464,29999221547154,110520245949639
%N A206273 Number of (n+1)X5 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206273 Column 4 of A206277
%H A206273 R. H. Hardin, <a href="/A206273/b206273.txt">Table of n, a(n) for n = 1..210</a>
%F A206273 Empirical: a(n) = 16*a(n-2) +17*a(n-4) -718*a(n-6) -808*a(n-8) +190*a(n-9) +11438*a(n-10) -620*a(n-11) -67647*a(n-12) +3492*a(n-13) +312771*a(n-14) -33638*a(n-15) +1348688*a(n-16) -102062*a(n-17) -853413*a(n-18) -750694*a(n-19) -7005863*a(n-20) -5658450*a(n-21) -25736013*a(n-22) +2139694*a(n-23) +41772730*a(n-24) -53739682*a(n-25) -164352002*a(n-26) +21736512*a(n-27) -2519634*a(n-28) -337012654*a(n-29) -81817424*a(n-30) +250893326*a(n-31) +917258192*a(n-32) -1447771298*a(n-33) -3615704782*a(n-34) -244892440*a(n-35) -7691321895*a(n-36) -2662626796*a(n-37) -12292738880*a(n-38) -2993682446*a(n-39) -10162211538*a(n-40) -7852874164*a(n-41) -31856358785*a(n-42) -200802094*a(n-43) -45550537463*a(n-44) -8070319668*a(n-45) -40592892197*a(n-46) -7781538872*a(n-47) -2259711479*a(n-48) -4398359126*a(n-49) +6341638113*a(n-50) +13027313724*a(n-51) +5351561306*a(n-52) +11253724846*a(n-53) -5223909939*a(n-54) +3668371886*a(n-55) +2746211621*a(n-56) +967459284*a(n-57) +5059156578*a(n-58) +4482864880*a(n-59) +3864467452*a(n-60) +3432722708*a(n-61) +1699749100*a(n-62) +1983595068*a(n-63) +1420608808*a(n-64) +463076952*a(n-65) +968625564*a(n-66) +162765888*a(n-67) +473812528*a(n-68) +71231744*a(n-69) +153455092*a(n-70) +17059712*a(n-71) +38080432*a(n-72) -2547072*a(n-73) +6152688*a(n-74) -676832*a(n-76) for n>82
%e A206273 Some solutions for n=4
%e A206273 ..1..1..2..2..1....2..1..2..0..1....0..2..1..2..2....0..1..2..0..2
%e A206273 ..0..2..1..1..2....2..2..0..0..2....0..0..2..2..2....0..1..0..2..2
%e A206273 ..1..0..1..1..2....2..2..2..1..0....0..0..2..2..0....0..2..1..2..2
%e A206273 ..2..0..0..2..1....1..1..0..2..2....1..1..0..0..1....1..0..1..1..0
%e A206273 ..2..2..1..0..1....0..2..1..1..1....1..1..0..0..0....2..0..0..2..1
%K A206273 nonn,new
%O A206273 1,1
%A A206273 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206272
%S A206272 1233,3168,12429,39306,133866,437340,1528920,4972464,17378730,
%T A206272 56812524,197823117,647998686,2252305758,7394389266,25660071480,
%U A206272 84368598492,292322882925,962789033730,3330544867176,10985542604052,37947971462754
%N A206272 Number of (n+1)X4 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206272 Column 3 of A206277
%H A206272 R. H. Hardin, <a href="/A206272/b206272.txt">Table of n, a(n) for n = 1..210</a>
%F A206272 Empirical: a(n) = 12*a(n-2) +22*a(n-4) +2*a(n-5) -340*a(n-6) +44*a(n-7) -30*a(n-8) +302*a(n-9) -3011*a(n-10) +198*a(n-11) +10251*a(n-12) -5274*a(n-13) +71024*a(n-14) -31264*a(n-15) -65095*a(n-16) +356*a(n-17) -100260*a(n-18) -55796*a(n-19) +285507*a(n-20) -129166*a(n-21) -253871*a(n-22) +145844*a(n-23) -497761*a(n-24) -139938*a(n-25) +558865*a(n-26) -259172*a(n-27) +120937*a(n-28) -90988*a(n-29) -849840*a(n-30) -208948*a(n-31) -309854*a(n-32) +4632*a(n-33) +119043*a(n-34) +74848*a(n-35) +43437*a(n-36) +90324*a(n-37) +60444*a(n-38) +22592*a(n-39) +12708*a(n-40) +4144*a(n-41) +8976*a(n-42) +1024*a(n-43) +1024*a(n-44) for n>49
%e A206272 Some solutions for n=4
%e A206272 ..0..0..1..2....1..0..1..0....2..1..1..0....0..2..2..1....2..2..0..2
%e A206272 ..1..1..2..2....2..0..0..2....2..0..0..2....2..2..2..1....2..1..2..2
%e A206272 ..1..1..2..2....2..0..0..0....2..0..0..2....2..2..0..2....1..2..2..2
%e A206272 ..2..2..0..0....1..2..0..0....0..1..1..0....0..0..1..1....2..2..2..0
%e A206272 ..2..2..2..1....1..1..2..2....0..1..1..0....0..0..1..1....2..2..0..1
%K A206272 nonn,new
%O A206272 1,1
%A A206272 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206271
%S A206271 306,972,3168,11205,35922,123357,408198,1376754,4572942,15438600,
%T A206271 51367338,172747923,576448980,1935886518,6465571452,21695411226,
%U A206271 72514916274,243164479179,813186470136,2725748659476,9118451550024
%N A206271 Number of (n+1)X3 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206271 Column 2 of A206277
%H A206271 R. H. Hardin, <a href="/A206271/b206271.txt">Table of n, a(n) for n = 1..210</a>
%F A206271 Empirical: a(n) = 8*a(n-2) +28*a(n-4) +36*a(n-5) -18*a(n-6) +78*a(n-7) -490*a(n-8) +298*a(n-9) +86*a(n-10) -854*a(n-11) +1669*a(n-12) -524*a(n-13) -872*a(n-14) +648*a(n-15) +140*a(n-16) -302*a(n-17) +10*a(n-18) +76*a(n-19) +8*a(n-20) -16*a(n-21) +8*a(n-22) for n>24
%e A206271 Some solutions for n=4
%e A206271 ..2..0..2....1..2..0....0..1..1....0..2..2....2..1..0....0..0..0....1..2..1
%e A206271 ..1..1..0....1..0..1....2..1..1....0..1..2....2..2..1....1..2..0....0..1..1
%e A206271 ..1..1..0....2..1..1....0..2..2....0..0..1....2..2..2....1..2..1....0..0..2
%e A206271 ..1..2..1....2..1..1....0..2..2....0..0..1....0..2..2....0..0..2....2..1..0
%e A206271 ..0..1..1....0..2..1....1..0..2....2..1..2....2..0..2....0..0..1....1..0..0
%K A206271 nonn,new
%O A206271 1,1
%A A206271 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206270
%S A206270 81,306,1233,4884,19509,77580,309057,1230480,4900461,19512618,
%T A206270 77701857,309412494,1232105853,4906303206,19537226541,77798456808,
%U A206270 309798343185,1233636035952,4912416015885,19561547012604,77895299289393
%N A206270 Number of (n+1)X2 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206270 Column 1 of A206277
%H A206270 R. H. Hardin, <a href="/A206270/b206270.txt">Table of n, a(n) for n = 1..210</a>
%F A206270 Empirical: a(n) = 6*a(n-2) +20*a(n-3) +58*a(n-4) +66*a(n-5) +29*a(n-6) +16*a(n-7)
%e A206270 Some solutions for n=4
%e A206270 ..1..0....2..1....0..1....1..1....2..1....2..0....1..1....2..2....2..1....1..1
%e A206270 ..0..2....2..0....2..0....1..0....1..1....1..2....0..1....1..2....0..2....1..2
%e A206270 ..1..1....2..0....2..2....2..1....1..1....1..2....2..1....0..2....1..2....1..0
%e A206270 ..2..0....2..1....1..0....2..2....0..1....1..0....0..2....2..0....2..2....2..1
%e A206270 ..0..0....0..2....2..1....2..2....2..2....2..1....2..2....0..0....1..0....1..1
%K A206270 nonn,new
%O A206270 1,1
%A A206270 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206269
%S A206269 81,972,12429,145824,2243139,39044748,964315869,27362301828,
%T A206269 1106854279383,53489846569626,3690406822607805
%N A206269 Number of (n+1)X(n+1) 0..2 arrays with no 2X2 subblock having the number of clockwise edge increases equal to the number of anticlockwise edge increases in its adjacent leftward or upward neighbors
%C A206269 Diagonal of A206277
%e A206269 Some solutions for n=4
%e A206269 ..2..1..0..2..1....2..0..0..2..1....0..2..2..0..1....1..1..1..0..2
%e A206269 ..0..1..1..0..0....2..0..0..0..2....0..1..2..1..0....1..1..2..1..1
%e A206269 ..0..1..1..0..0....0..2..0..0..2....1..1..0..2..2....0..2..0..0..1
%e A206269 ..1..0..0..1..0....0..0..1..1..0....0..2..1..2..0....2..0..0..0..2
%e A206269 ..2..0..0..0..2....0..0..1..1..2....1..0..1..1..0....2..0..0..1..0
%K A206269 nonn,new
%O A206269 1,1
%A A206269 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206267
%S A206267 16,24,24,34,28,34,50,44,44,50,72,64,78,64,72,105,93,135,135,93,105,
%T A206267 154,128,218,260,218,128,154,229,174,339,471,471,339,174,229,345,228,
%U A206267 503,800,932,800,503,228,345,527,295,724,1296,1725,1725,1296,724,295,527,815,372
%N A206267 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206267 Table starts
%C A206267 ..16..24..34...50...72...105...154...229...345....527....815...1274...2009
%C A206267 ..24..28..44...64...93...128...174...228...295....372....464....568....689
%C A206267 ..34..44..78..135..218...339...503...724..1009...1374...1828...2389...3068
%C A206267 ..50..64.135..260..471...800..1296..2010..3012...4376...6197...8576..11637
%C A206267 ..72..93.218..471..932..1725..3011..5014..8016..12385..18572..27141..38768
%C A206267 .105.128.339..800.1725..3440..6444.11448.19457..31832..50397..77528.116289
%C A206267 .154.174.503.1296.3011..6444.12878.24319.43766..75591.125978.203499.319778
%C A206267 .229.228.724.2010.5014.11448.24319.48628.92387.167968.293939.497428.817199
%H A206267 R. H. Hardin, <a href="/A206267/b206267.txt">Table of n, a(n) for n = 1..7894</a>
%F A206267 Empirical for column k:
%F A206267 k=1: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
%F A206267 k=2: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) for n>6
%F A206267 k=3: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6) for n>7
%F A206267 k=4: a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7) for n>8
%F A206267 k=5: a(n) = 6*a(n-1) -14*a(n-2) +14*a(n-3) -14*a(n-5) +14*a(n-6) -6*a(n-7) +a(n-8) for n>9
%F A206267 k=6: a(n) = 7*a(n-1) -20*a(n-2) +28*a(n-3) -14*a(n-4) -14*a(n-5) +28*a(n-6) -20*a(n-7) +7*a(n-8) -a(n-9) for n>10
%F A206267 k=7: a(n) = 8*a(n-1) -27*a(n-2) +48*a(n-3) -42*a(n-4) +42*a(n-6) -48*a(n-7) +27*a(n-8) -8*a(n-9) +a(n-10) for n>11
%e A206267 Some solutions for n=4 k=3
%e A206267 ..1..0..0..0....0..0..0..0....1..1..0..0....0..0..1..0....1..0..1..0
%e A206267 ..0..0..0..0....0..0..0..0....1..1..0..0....0..1..0..1....0..1..0..1
%e A206267 ..0..0..0..0....0..0..0..0....0..0..0..0....1..0..1..0....1..0..1..0
%e A206267 ..0..0..0..0....0..0..0..0....0..0..0..0....0..1..0..1....0..1..0..1
%e A206267 ..0..0..0..0....0..0..0..0....0..0..0..0....1..0..1..1....1..0..1..1
%K A206267 nonn,tabl,new
%O A206267 1,1
%A A206267 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206266
%S A206266 154,174,503,1296,3011,6444,12878,24319,43766,75591,125978,203499,
%T A206266 319778,490323,735479,1081584,1562283,2220084,3108113,4292154,5852933,
%U A206266 7888734,10518308,13884165,18156212,23535829,30260348,38608029,48903500,61523757
%N A206266 Number of (n+1)X8 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206266 Column 7 of A206267
%H A206266 R. H. Hardin, <a href="/A206266/b206266.txt">Table of n, a(n) for n = 1..210</a>
%F A206266 Empirical: a(n) = 8*a(n-1) -27*a(n-2) +48*a(n-3) -42*a(n-4) +42*a(n-6) -48*a(n-7) +27*a(n-8) -8*a(n-9) +a(n-10) for n>11
%e A206266 Some solutions for n=4
%e A206266 ..0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0....1..1..1..1..0..0..0..0
%e A206266 ..0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0....1..1..1..1..0..0..0..0
%e A206266 ..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0
%e A206266 ..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0
%e A206266 ..0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....1..0..0..0..0..0..0..0
%K A206266 nonn,new
%O A206266 1,1
%A A206266 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206265
%S A206265 105,128,339,800,1725,3440,6444,11448,19457,31832,50397,77528,116289,
%T A206265 170552,245166,346112,480709,657808,888039,1184048,1560789,2035808,
%U A206265 2629584,3365864,4272057,5379624,6724529,8347688,10295481,12620264
%N A206265 Number of (n+1)X7 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206265 Column 6 of A206267
%H A206265 R. H. Hardin, <a href="/A206265/b206265.txt">Table of n, a(n) for n = 1..210</a>
%F A206265 Empirical: a(n) = 7*a(n-1) -20*a(n-2) +28*a(n-3) -14*a(n-4) -14*a(n-5) +28*a(n-6) -20*a(n-7) +7*a(n-8) -a(n-9) for n>10
%e A206265 Some solutions for n=4
%e A206265 ..0..1..0..1..0..1..0....1..0..0..1..1..0..0....0..1..0..1..0..1..0
%e A206265 ..1..0..1..0..1..0..1....1..1..0..0..1..1..0....1..0..1..0..1..0..1
%e A206265 ..0..1..0..1..0..1..0....0..1..1..0..0..1..1....0..1..0..1..0..1..0
%e A206265 ..1..0..1..0..1..0..1....0..0..1..1..0..0..1....1..0..1..0..1..0..1
%e A206265 ..0..1..0..1..0..1..0....1..0..0..1..1..0..0....0..1..0..1..0..1..1
%K A206265 nonn,new
%O A206265 1,1
%A A206265 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206264
%S A206264 72,93,218,471,932,1725,3011,5014,8016,12385,18572,27141,38768,54273,
%T A206264 74621,100956,134604,177109,230238,296019,376748,475029,593783,736290,
%U A206264 906200,1107577,1344912,1623169,1947800,2324793,2760689,3262632,3838388
%N A206264 Number of (n+1)X6 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206264 Column 5 of A206267
%H A206264 R. H. Hardin, <a href="/A206264/b206264.txt">Table of n, a(n) for n = 1..210</a>
%F A206264 Empirical: a(n) = 6*a(n-1) -14*a(n-2) +14*a(n-3) -14*a(n-5) +14*a(n-6) -6*a(n-7) +a(n-8) for n>9
%e A206264 Some solutions for n=4
%e A206264 ..1..0..1..0..1..0....1..1..1..1..0..0....0..1..0..1..0..1....0..0..0..0..0..0
%e A206264 ..0..1..0..1..0..1....0..0..0..0..0..0....1..0..1..0..1..0....0..0..0..0..0..0
%e A206264 ..1..0..1..0..1..0....0..0..0..0..0..0....0..1..0..1..0..1....0..0..0..0..0..0
%e A206264 ..0..1..0..1..0..1....0..0..0..0..0..0....1..0..1..0..1..0....0..0..0..0..0..0
%e A206264 ..1..0..1..0..1..1....0..0..0..0..0..0....0..1..0..1..0..1....0..0..0..0..0..0
%K A206264 nonn,new
%O A206264 1,1
%A A206264 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206263
%S A206263 50,64,135,260,471,800,1296,2010,3012,4376,6197,8576,11637,15512,
%T A206263 20358,26342,33658,42512,53139,65788,80739,98288,118764,142514,169920,
%U A206263 201384,237345,278264,324641,377000,435906,501950,575766,658016,749407,850676
%N A206263 Number of (n+1)X5 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206263 Column 4 of A206267
%H A206263 R. H. Hardin, <a href="/A206263/b206263.txt">Table of n, a(n) for n = 1..210</a>
%F A206263 Empirical: a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7) for n>8
%e A206263 Some solutions for n=4
%e A206263 ..1..1..0..0..1....1..1..0..0..0....1..1..1..1..0....1..0..0..0..0
%e A206263 ..0..1..1..0..0....1..0..0..0..0....0..0..0..0..0....1..0..0..0..0
%e A206263 ..0..0..1..1..0....1..0..0..0..0....0..0..0..0..0....1..0..0..0..0
%e A206263 ..1..0..0..1..1....0..0..0..0..0....0..0..0..0..0....1..0..0..0..0
%e A206263 ..1..1..0..0..1....0..0..0..0..0....0..0..0..0..0....1..0..0..0..0
%K A206263 nonn,new
%O A206263 1,1
%A A206263 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206262
%S A206262 34,44,78,135,218,339,503,724,1009,1374,1828,2389,3068,3885,4853,5994,
%T A206262 7323,8864,10634,12659,14958,17559,20483,23760,27413,31474,35968,
%U A206262 40929,46384,52369,58913,66054,73823,82260,91398,101279,111938,123419,135759
%N A206262 Number of (n+1)X4 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206262 Column 3 of A206267
%H A206262 R. H. Hardin, <a href="/A206262/b206262.txt">Table of n, a(n) for n = 1..210</a>
%F A206262 Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6) for n>7
%e A206262 Some solutions for n=4
%e A206262 ..1..1..1..1....0..0..1..1....1..1..1..1....1..1..0..0....0..0..1..0
%e A206262 ..0..0..0..0....1..0..0..1....1..1..1..1....1..1..0..0....0..1..0..1
%e A206262 ..0..0..0..0....1..1..0..0....1..1..0..0....0..0..0..0....1..0..1..0
%e A206262 ..0..0..0..0....0..1..1..0....0..0..0..0....0..0..0..0....0..1..0..1
%e A206262 ..0..0..0..0....0..0..1..1....0..0..0..0....0..0..0..0....1..0..1..1
%K A206262 nonn,new
%O A206262 1,1
%A A206262 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206261
%S A206261 24,28,44,64,93,128,174,228,295,372,464,568,689,824,978,1148,1339,
%T A206261 1548,1780,2032,2309,2608,2934,3284,3663,4068,4504,4968,5465,5992,
%U A206261 6554,7148,7779,8444,9148,9888,10669,11488,12350,13252,14199,15188,16224,17304
%N A206261 Number of (n+1)X3 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206261 Column 2 of A206267
%H A206261 R. H. Hardin, <a href="/A206261/b206261.txt">Table of n, a(n) for n = 1..210</a>
%F A206261 Empirical: a(n) = 3*a(n-1) -2*a(n-2) -2*a(n-3) +3*a(n-4) -a(n-5) for n>6
%e A206261 Some solutions for n=4
%e A206261 ..1..1..0....1..0..1....1..0..0....0..1..1....1..0..0....0..0..1....1..1..1
%e A206261 ..0..1..1....0..1..0....1..1..0....0..0..1....0..0..0....0..1..0....1..1..1
%e A206261 ..0..0..1....1..0..1....0..1..1....1..0..0....0..0..0....1..0..1....1..1..0
%e A206261 ..1..0..0....0..1..0....0..0..1....1..1..0....0..0..0....0..1..0....1..0..0
%e A206261 ..1..1..0....1..0..1....1..0..0....0..1..1....0..0..0....1..0..1....1..0..0
%K A206261 nonn,new
%O A206261 1,1
%A A206261 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206260
%S A206260 16,24,34,50,72,105,154,229,345,527,815,1274,2009,3190,5092,8160,
%T A206260 13114,21119,34060,54987,88835,143589,232169,375480,607347,982500,
%U A206260 1589494,2571614,4160700,6731877,10892110,17623489,28515069,46137995,74652467
%N A206260 Number of (n+1)X2 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206260 Column 1 of A206267
%H A206260 R. H. Hardin, <a href="/A206260/b206260.txt">Table of n, a(n) for n = 1..210</a>
%F A206260 Empirical: a(n) = 4*a(n-1) -5*a(n-2) +a(n-3) +2*a(n-4) -a(n-5) for n>6
%e A206260 Some solutions for n=4
%e A206260 ..0..0....0..1....0..1....1..1....1..0....1..1....0..0....1..1....0..0....1..0
%e A206260 ..0..1....1..0....1..0....0..0....0..0....1..1....1..0....1..0....0..0....1..1
%e A206260 ..1..0....0..1....0..1....0..0....0..0....1..1....1..1....0..0....0..0....0..1
%e A206260 ..0..1....1..0....0..1....0..0....0..0....1..0....0..1....0..0....0..0....0..0
%e A206260 ..1..0....0..1....1..0....0..0....0..0....1..0....0..0....0..0....0..0....1..0
%K A206260 nonn,new
%O A206260 1,1
%A A206260 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206259
%S A206259 16,28,78,260,932,3440,12878,48628,184764,705440,2704164,10400608,
%T A206259 40116608,155117528,601080398,2333606228,9075135308,35345263808,
%U A206259 137846528828,538257874448,2104098963728,8233430727608,32247603683108
%N A206259 Number of (n+1)X(n+1) 0..1 arrays with the number of rightwards and downwards edge increases in each 2X2 subblock equal to the number in all its horizontal and vertical neighbors
%C A206259 Diagonal of A206267
%H A206259 R. H. Hardin, <a href="/A206259/b206259.txt">Table of n, a(n) for n = 1..69</a>
%e A206259 Some solutions for n=4
%e A206259 ..1..1..0..0..0....0..1..0..1..0....1..1..1..1..0....1..1..0..0..0
%e A206259 ..1..0..0..0..0....1..0..1..0..1....0..0..0..0..0....1..0..0..0..0
%e A206259 ..0..0..0..0..0....0..1..0..1..0....0..0..0..0..0....1..0..0..0..0
%e A206259 ..0..0..0..0..0....1..0..1..0..1....0..0..0..0..0....0..0..0..0..0
%e A206259 ..0..0..0..0..0....0..1..0..1..0....0..0..0..0..0....0..0..0..0..0
%K A206259 nonn,new
%O A206259 1,1
%A A206259 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A195308
%S A195308 1,1,1,2,3,5,7,11,15,22,30,43,58,81,109,150,200,271,359,481,633,838,
%T A195308 1095,1438,1867,2430,3136,4053,5200,6676,8519
%N A195308 a(n) = A005291(n) + A005291(n+1).
%C A195308 This sequence arises from A005291 in the same way as A000041 arises from A182712.
%C A195308 Observation: a(3)..a(13) coincide with a sequence related to Stirling's numbers from the Jordan's book.
%D A195308 C. Jordan, Calculus of finite differences, Chelsea Publishing Co. (1965), chapter IV, 153-155.
%Y A195308 Cf. A000041, A005291, A087787, A100818, A182712, A194439, A195017.
%K A195308 nonn,more,new
%O A195308 1,4
%A A195308 Omar E. Pol (info(AT)polprimos.com), Feb 03 2012

%I A206258
%S A206258 5,13,29,45,77,93,141,173,221,253,333,365,461,509,573,637,765,813,957,
%T A206258 1021,1117,1197,1373,1437,1597,1693,1837,1933,2157,2221,2461,2589,
%U A206258 2749,2877,3069,3165,3453,3597,3789,3917,4237,4333,4669,4829,5021,5197,5565
%N A206258 1/8 the number of 2X2 -n..n arrays with a 2X2 -n..n inverse, i.e. with determinant +-1
%H A206258 R. H. Hardin, <a href="/A206258/b206258.txt">Table of n, a(n) for n = 1..334</a>
%F A206258 a(n) = a(n-1) +8*A000010(n)
%K A206258 nonn,new
%O A206258 1,1
%A A206258 R. H. Hardin (rhhardin(AT)att.net), with Max Alekseyev (maxale(AT)gmail.com) in the Sequence Fans Mailing List, Feb 05 2012

%I A206255
%S A206255 49,361,361,1600,8029,1600,9409,99856,99856,9409,47089,1718209,
%T A206255 1364224,1718209,47089,258064,26512201,49336576,49336576,26512201,
%U A206255 258064,1343281,434613664,944701696,4556324929,944701696,434613664,1343281,7198489
%N A206255 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having zero permanent
%C A206255 Table starts
%C A206255 ......49..........361...........1600..............9409................47089
%C A206255 .....361.........8029..........99856...........1718209.............26512201
%C A206255 ....1600........99856........1364224..........49336576............944701696
%C A206255 ....9409......1718209.......49336576........4556324929.........226637884225
%C A206255 ...47089.....26512201......944701696......226637884225.......14139111560401
%C A206255 ..258064....434613664....27285753856....16533087493120.....2261195747329024
%C A206255 .1343281...6990799321...603269103616..1006239668280961...186370092021300625
%C A206255 .7198489.113636628469.15860529270784.68438717405988481.24455901146017548025
%H A206255 R. H. Hardin, <a href="/A206255/b206255.txt">Table of n, a(n) for n = 1..611</a>
%F A206255 Column 1 is A006130(n+2)^2
%e A206255 Some solutions for n=4 k=3
%e A206255 ..2..0..3..3....0..0..1..1....0..3..3..0....0..0..0..0....0..2..0..3
%e A206255 ..0..0..0..0....3..0..0..0....0..0..0..0....3..0..1..1....0..0..0..1
%e A206255 ..2..1..1..0....3..0..1..1....1..0..1..0....2..0..0..0....1..0..0..1
%e A206255 ..0..0..0..0....1..0..0..0....1..0..0..0....3..0..2..0....0..0..0..3
%e A206255 ..2..1..1..0....3..0..0..2....2..0..2..3....2..0..3..0....1..0..0..1
%K A206255 nonn,tabl,new
%O A206255 1,1
%A A206255 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206256
%S A206256 1,2,5,4,8,6,9,8,0,9,0,5,8
%N A206256 Decimal expansion of Prod_{p prime} (1 - 3/p^2).
%C A206256 This is the probability that n, n+1, n+2 all are squarefree.
%e A206256 0.1254869809058...
%Y A206256 Cf. A059956.
%K A206256 nonn,cons,more,new
%O A206256 0,2
%A A206256 N. J. A. Sloane (njas(AT)research.att.com), Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012

%I A206254
%S A206254 1343281,6990799321,603269103616,1006239668280961,186370092021300625,
%T A206254 173223627538782109696,48673249000646115009649,
%U A206254 33073672641937525219176361,11681343800269067725781401600
%N A206254 Number of (n+1)X8 0..3 arrays with every 2X2 subblock having zero permanent
%C A206254 Column 7 of A206255
%H A206254 R. H. Hardin, <a href="/A206254/b206254.txt">Table of n, a(n) for n = 1..210</a>
%F A206254 Empirical: a(n) = 760*a(n-1) +128388*a(n-2) -213382620*a(n-3) +16145776764*a(n-4) +22629583010268*a(n-5) -3838103572007826*a(n-6) -1210315733012118204*a(n-7) +300718177544003494572*a(n-8) +34602416054049234499380*a(n-9) -12994232799115556406238428*a(n-10) -424461018655040288964274332*a(n-11) +354738088375016440542715465857*a(n-12) -4468102499760252335669263322700*a(n-13) -6503279595087148111920138775297704*a(n-14) +277510620511494531092021166490945608*a(n-15) +82474473929335317230775863351392167768*a(n-16) -5498152145270648546778141658473380210280*a(n-17) -730250556612024541625665464361980553987644*a(n-18) +66038827852503719715205962805391055657831528*a(n-19) +4454491535522432110676326644177520535672544216*a(n-20) -540108571898317929022757321728674058151654948856*a(n-21) -17638208163687534281555446585669375079024593370040*a(n-22) +3137561439154039104522506690131458161394475752281160*a(n-23) +35177741666224179894440470136683380207557950404881265*a(n-24) -13167077208904047000367865266944534320529756493094436000*a(n-25) +43688889198602094578451687709368723846189065605963706100*a(n-26) +40059681408274406058034945112761538656935934971228508218900*a(n-27) -559303998260830692987736964847796546788703062392908431172500*a(n-28) -87771432361178289178636901099484386130605650745628635269864500*a(n-29) +1970952230186641634810668603114964552917890758676216330487800750*a(n-30) +136077400810169689982526569739166662423371010718064025496825012500*a(n-31) -4095902546984185377950722262594545453140105402446965117499282482500*a(n-32) -144310246460892961724788287171396935379985513568288457266808382717500*a(n-33) +5542901101468143355388555017417245989331094924883170007175870876412500*a(n-34) +97863154430359391429714370578878589638058204950929985039872979925812500*a(n-35) -4948606276747342422592839157088698145520612955481243300239160619474140625*a(n-36) -35626692188290247005328357918844203986310354588928055050257115648589687500*a(n-37) +2846147567503871277417727284205987598885268367591282809508000271004309750000*a(n-38) +1558504496045032163427441148226905337799471898023015748036005485495274000000*a(n-39) -994139983023571208552171600344573741307284156355708460656826831169915800000000*a(n-40) +3790457841254650180660547098562506369032591860002027197728771672226933600000000*a(n-41) +186441087666122050344500007570415430891032704492688753507221117662492164800000000*a(n-42) -1238842193344667836517765745100794188644120350672317241137923372915102560000000000*a(n-43) -13707763404230765904256670929061150807489951650788484448276534272953628800000000000*a(n-44) +112820113558796538181052142390479967845703559143834940276767811044190218240000000000*a(n-45)
%e A206254 Some solutions for n=4
%e A206254 ..1..0..0..2..3..0..1..1....2..2..3..1..3..3..0..0....3..1..3..1..3..3..3..1
%e A206254 ..1..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0....0..0..0..0..0..0..0..0
%e A206254 ..3..0..2..2..1..1..2..1....3..3..1..2..0..3..0..1....1..2..3..0..1..2..0..0
%e A206254 ..0..0..0..0..0..0..0..0....0..0..0..0..0..2..0..0....0..0..0..0..0..0..0..2
%e A206254 ..3..2..2..2..3..1..2..0....0..3..0..3..0..2..0..3....0..2..0..1..0..3..0..2
%K A206254 nonn,new
%O A206254 1,1
%A A206254 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206253
%S A206253 258064,434613664,27285753856,16533087493120,2261195747329024,
%T A206253 848170637644693504,173223627538782109696,53060397281015908925440,
%U A206253 12918833834289527406198784,3660962109260466462309744640
%N A206253 Number of (n+1)X7 0..3 arrays with every 2X2 subblock having zero permanent
%C A206253 Column 6 of A206255
%H A206253 R. H. Hardin, <a href="/A206253/b206253.txt">Table of n, a(n) for n = 1..210</a>
%F A206253 Empirical: a(n) = 436*a(n-1) -20670912*a(n-3) +1768362192*a(n-4) +301989632832*a(n-5) -40539456253440*a(n-6) -1536342178669056*a(n-7) +377148789279765504*a(n-8) -1615023812548104192*a(n-9) -1758635101880092164096*a(n-10) +45109122367823348465664*a(n-11) +4303432979017649071128576*a(n-12) -181303558019449768187265024*a(n-13) -5085618071625131560739536896*a(n-14) +338719378543908534036612513792*a(n-15) +1447054855782983096105476030464*a(n-16) -323240313366898115796464692101120*a(n-17) +2430629740180983250878668715589632*a(n-18) +151402677346160235585093709301022720*a(n-19) -2307461851889299859661071648393527296*a(n-20) -28010992393774322798242500381933305856*a(n-21) +696331835357100763665844862341317918720*a(n-22) -69186082604013077340806446462997605908480*a(n-24) +353941433111056374606651926115966699700224*a(n-25)
%e A206253 Some solutions for n=4
%e A206253 ..3..1..0..0..1..3..0....1..1..3..2..3..2..0....3..0..0..1..0..2..0
%e A206253 ..0..0..0..0..0..0..0....0..0..0..0..0..0..0....1..0..0..1..0..1..0
%e A206253 ..2..3..1..0..3..2..0....3..2..3..0..1..0..2....2..0..0..3..0..0..0
%e A206253 ..0..0..0..0..0..0..0....0..0..0..0..2..0..0....3..0..0..2..0..1..2
%e A206253 ..1..0..3..1..0..1..2....1..1..1..0..1..0..2....0..0..0..2..0..0..0
%K A206253 nonn,new
%O A206253 1,1
%A A206253 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206252
%S A206252 47089,26512201,944701696,226637884225,14139111560401,
%T A206252 2261195747329024,186370092021300625,24455901146017548025,
%U A206252 2314068144009520254976,275508832683852646545601,27911391658467202945715041
%N A206252 Number of (n+1)X6 0..3 arrays with every 2X2 subblock having zero permanent
%C A206252 Column 5 of A206255
%H A206252 R. H. Hardin, <a href="/A206252/b206252.txt">Table of n, a(n) for n = 1..210</a>
%F A206252 Empirical: a(n) = 133*a(n-1) +6888*a(n-2) -1340811*a(n-3) +4226391*a(n-4) +4180767372*a(n-5) -80490770091*a(n-6) -5217648903441*a(n-7) +148692944523528*a(n-8) +2437871310147447*a(n-9) -99376895146018923*a(n-10) -199651087166898780*a(n-11) +26256479575727404800*a(n-12) -103833901082736072000*a(n-13) -2123373038352913440000*a(n-14) +13177032454057536000000*a(n-15)
%e A206252 Some solutions for n=4
%e A206252 ..1..2..2..0..3..2....3..2..3..1..1..2....2..2..2..2..3..2....1..3..3..1..2..0
%e A206252 ..0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0
%e A206252 ..0..0..2..3..0..1....3..0..1..1..2..0....1..3..2..3..1..0....2..2..0..1..3..2
%e A206252 ..1..0..0..0..0..1....2..0..0..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0
%e A206252 ..0..0..1..0..0..2....3..0..0..1..2..3....2..3..0..3..3..0....1..3..3..3..2..3
%K A206252 nonn,new
%O A206252 1,1
%A A206252 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206251
%S A206251 9409,1718209,49336576,4556324929,226637884225,16533087493120,
%T A206251 1006239668280961,68438717405988481,4427204221759169536,
%U A206251 295306870252087238017,19431114832514090805121,1289257317404589497597440
%N A206251 Number of (n+1)X5 0..3 arrays with every 2X2 subblock having zero permanent
%C A206251 Column 4 of A206255
%H A206251 R. H. Hardin, <a href="/A206251/b206251.txt">Table of n, a(n) for n = 1..210</a>
%F A206251 Empirical: a(n) = 88*a(n-1) -128439*a(n-3) +1879848*a(n-4) +28773144*a(n-5) -593002863*a(n-6) +32230833768*a(n-8) -98419153113*a(n-9)
%e A206251 Some solutions for n=4
%e A206251 ..3..3..2..3..0....3..1..1..2..0....0..1..0..2..3....2..1..3..3..1
%e A206251 ..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
%e A206251 ..3..0..0..0..3....1..2..1..1..3....0..0..3..2..3....0..1..0..2..2
%e A206251 ..2..0..2..0..3....0..0..0..0..0....1..0..0..0..0....0..1..0..0..0
%e A206251 ..3..0..3..0..0....2..1..1..2..1....0..0..0..1..1....0..1..0..1..3
%K A206251 nonn,new
%O A206251 1,1
%A A206251 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206250
%S A206250 1600,99856,1364224,49336576,944701696,27285753856,603269103616,
%T A206250 15860529270784,372641945092096,9417680785309696,227009527113908224,
%U A206250 5642600260896292864,137487696199369621504,3393605721164360974336
%N A206250 Number of (n+1)X4 0..3 arrays with every 2X2 subblock having zero permanent
%C A206250 Column 3 of A206255
%H A206250 R. H. Hardin, <a href="/A206250/b206250.txt">Table of n, a(n) for n = 1..210</a>
%F A206250 Empirical: a(n) = 28*a(n-1) +192*a(n-2) -8352*a(n-3) +25920*a(n-4) +373248*a(n-5) -1679616*a(n-6)
%e A206250 Some solutions for n=4
%e A206250 ..1..0..0..1....2..0..3..3....2..1..2..0....0..0..1..1....1..0..0..0
%e A206250 ..0..0..0..1....0..0..0..0....0..0..0..0....3..0..0..0....0..0..1..2
%e A206250 ..1..0..0..0....2..1..1..0....2..0..1..3....3..0..1..1....1..0..0..0
%e A206250 ..1..0..0..3....0..0..0..0....2..0..0..0....1..0..0..0....1..0..3..0
%e A206250 ..2..0..0..1....2..1..1..0....0..0..0..1....3..0..0..2....1..0..0..0
%K A206250 nonn,new
%O A206250 1,1
%A A206250 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206249
%S A206249 361,8029,99856,1718209,26512201,434613664,6990799321,113636628469,
%T A206249 1841188465216,29885540089249,484822312979761,7867521981505504,
%U A206249 127659187514359081,2071519703855856109,33613908262858076176
%N A206249 Number of (n+1)X3 0..3 arrays with every 2X2 subblock having zero permanent
%C A206249 Column 2 of A206255
%H A206249 R. H. Hardin, <a href="/A206249/b206249.txt">Table of n, a(n) for n = 1..210</a>
%F A206249 Empirical: a(n) = 19*a(n-1) -855*a(n-3) +2025*a(n-4)
%e A206249 Some solutions for n=4
%e A206249 ..0..2..3....0..2..0....0..0..3....3..0..1....1..2..2....0..3..3....2..3..0
%e A206249 ..0..0..0....0..0..0....1..0..1....0..0..1....0..0..0....0..0..0....0..0..0
%e A206249 ..1..1..1....1..3..2....0..0..3....0..0..1....1..1..1....1..0..0....3..3..3
%e A206249 ..0..0..0....0..0..0....2..0..3....1..0..0....0..0..0....1..0..3....0..0..0
%e A206249 ..3..1..3....1..0..1....2..0..1....3..0..1....0..0..2....2..0..1....1..1..0
%K A206249 nonn,new
%O A206249 1,1
%A A206249 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206248
%S A206248 49,361,1600,9409,47089,258064,1343281,7198489,37945600,201895681,
%T A206248 1068570721,5672499856,30061664689,159465247561,845443470400,
%U A206248 4483691905729,23774527583569,126075439502224,668536407557041
%N A206248 Number of (n+1)X2 0..3 arrays with every 2X2 subblock having zero permanent
%C A206248 Column 1 of A206255
%H A206248 R. H. Hardin, <a href="/A206248/b206248.txt">Table of n, a(n) for n = 1..210</a>
%F A206248 Empirical: a(n) = 4*a(n-1) +12*a(n-2) -27*a(n-3)
%F A206248 a(n) = A006139(n+2)^2
%e A206248 Some solutions for n=4
%e A206248 ..1..0....2..1....2..1....2..3....2..3....3..3....3..3....0..3....0..3....0..1
%e A206248 ..0..0....0..0....0..0....0..0....0..0....0..0....0..0....0..2....0..2....0..0
%e A206248 ..0..3....1..0....2..1....2..3....0..0....1..1....3..1....0..3....0..0....0..1
%e A206248 ..0..1....2..0....0..0....0..0....0..0....0..0....0..0....0..2....3..3....0..2
%e A206248 ..0..3....1..0....3..0....1..3....3..0....1..1....0..1....0..1....0..0....0..2
%K A206248 nonn,new
%O A206248 1,1
%A A206248 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A184994
%S A184994 1,2,38,2018,210422,36297362,9356755718,3369557048258,
%T A184994 1615758952865942,995259055695876722,765831994417031276198,
%U A184994 719917951968845731560098,811830142106561351995390262
%N A184994 E.g.f. A(x)=sum(n>=0, a(n)*x^(2*n+1)/(2*n+1)! is inverse to f(x)=2*sin(x)-x
%F A184994 a(n)=2*((2*n)!*sum(k=1..2*n, binomial(2*n+k,2*n)*sum(j=1..k, binomial(k,j)*(sum(l=0.j-1, (binomial(j,l)*sum(i=0..(j-l)/2, binomial(j-l,i)*(l-j+2*i)^(2*n-l+j)*(-1)^(n-i)))/(2*n-l+j)!))))), a(0)=1.
%o A184994 (Maxima) a(n):=if n=0 then 1 else 2*((2*n)!*sum(binomial(2*n+k,2*n)*sum(binomial(k,j)*(sum((binomial(j,l)*sum(binomial(j-l,i)*(l-j+2*i)^(2*n-l+j)*(-1)^(n-i),i,0,(j-l)/2))/(2*n-l+j)!,l,0,j-1)),j,1,k),k,1,2*n));
%K A184994 nonn,new
%O A184994 0,2
%A A184994 Vladimir Kruchinin (kru(AT)ie.tusur.ru), Feb 04 2012

%I A206247
%S A206247 49,8029,1364224,4556324929,14139111560401,848170637644693504,
%T A206247 48673249000646115009649,50198078741005740094898080285,
%U A206247 52964434709981480280861081128206336
%N A206247 Number of (n+1)X(n+1) 0..3 arrays with every 2X2 subblock having zero permanent
%C A206247 Diagonal of A206255
%H A206247 R. H. Hardin, <a href="/A206247/b206247.txt">Table of n, a(n) for n = 1..17</a>
%e A206247 Some solutions for n=4
%e A206247 ..3..3..0..1..0....1..2..3..2..2....3..3..2..1..3....0..0..3..1..3
%e A206247 ..0..0..0..1..0....0..0..0..0..0....0..0..0..0..0....3..0..0..0..0
%e A206247 ..3..2..0..3..0....2..1..3..2..1....1..0..2..0..1....2..0..3..0..1
%e A206247 ..0..0..0..1..0....0..0..0..0..0....1..0..1..0..2....3..0..0..0..3
%e A206247 ..2..3..0..1..0....1..0..3..3..3....2..0..1..0..3....3..0..3..0..0
%K A206247 nonn,new
%O A206247 1,1
%A A206247 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A205959
%S A205959 1,1,1,2,1,6,1,4,3,10,1,24,1,14,15,8,1,54,1,40,21,22,1,96,5,26,9,56,1,
%T A205959 900,1,16,33,34,35,216,1,38,39,160,1,1764,1,88,135,46,1,384,7,250,51,
%U A205959 104,1,486,55,224,57,58,1,7200,1,62,189,32,65,4356,1,136
%N A205959 a(n) = exp(-Sum_{d in P} moebius(d)*log(n/d)) where P = {d : d divides n and d is prime}.
%C A205959 This is a variant of the (exponential of the) von Mangoldt function where the divisors are restricted to prime divisors. The (exponential of the) summatory function is A025527. Apart from n=1 the value is 1 if and only if n is prime; the fixed points are the products of two distinct primes (A006881).
%H A205959 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/SequenceTransformations">Transformations of Integer Sequences</a>.
%p A205959 with(numtheory): A205959 := proc(n) select(isprime, divisors(n));
%p A205959 simplify(exp(-add(mobius(d)*log(n/d), d=%))) end:
%p A205959 seq(A205959(i),i=1..60);
%o A205959 (Sage)
%o A205959 def A205959(n) :
%o A205959     P = filter(is_prime, divisors(n))
%o A205959     return simplify(exp(-add(moebius(d)*log(n/d) for d in P)))
%o A205959 [A205959(n) for n in (1..60)]
%Y A205959 Cf. A003418, A025527, A008578, A102467, A006881.
%K A205959 nonn,nice,new
%O A205959 1,4
%A A205959 Peter Luschny (peter(AT)luschny.de), Feb 03 2012

%I A206241
%S A206241 0,1,1,2,3,8,11,19,49,68,117,185,487,672,1159,1831,4821,6652,11473,
%T A206241 18125,47723,65848,113571,179419,472409,651828,1124237,1776065,
%U A206241 4676367,6452432,11128799,17581231,46291261,63872492,110163753,284199998,394363751,678563749,1751491249
%N A206241 a(n) is the smallest number of the form k*a(n-1)+a(n-2) for k>0 that is relatively prime to n, with a(0) = 0 and a(1) = 1.
%o A206241 (PARI) ar(n)=local(r,m);r=vector(n);r[1]=0;r[2]=1;for(k=2,n-1,m=1;while(gcd(m*r[k]+r[k-1],k)>1,m++);r[k+1]=m*r[k]+r[k-1]);r
%K A206241 nonn,new
%O A206241 0,4
%A A206241 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Feb 05 2012

%I A185190
%S A185190 1,2,12,112,1440,23648,473088,11164032,303624960,9351301632,
%T A185190 321717276672,12228424826880,508916576243712,23016333612318720,
%U A185190 1124014843389984768,58949533609403842560,3304473379374295744512,197167421810210663301120,12476358616574849161101312
%N A185190 E.g.f. satisfies: A(x) = x + asinh(A(x))^2.
%C A185190 Radius of convergence of the e.g.f. A(x) is r = 0.2767308231516982...,
%C A185190 where r and A(r) satisfy: r = A(r) - (1 + A(r)^2)/4 and
%C A185190 A(r) = sinh( sqrt(1 + A(r)^2)/2 ), so that A(r) = 0.6241087588791013...
%F A185190  E.g.f.: A(x) = Series_Reversion( x - asinh(x)^2 ).
%F A185190 E.g.f. derivative: A'(x) = 1/(1 - 2*asinh(A(x))/sqrt(1 + A(x)^2) ).
%e A185190 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%e A185190 Related expansions:
%e A185190 asinh(A(x)) = x + 2*x^2/2! + 11*x^3/3! + 100*x^4/4! + 1269*x^5/5! +...
%e A185190 asinh(A(x))^2 =  2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! +...
%o A185190 (PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + asinh(A)^2); n!*polcoeff(A, n)}
%o A185190 (PARI) {a(n)=n!*polcoeff(serreverse(x-asinh(x+x*O(x^n))^2), n)}
%o A185190 for(n=1,26,print1(a(n),","))
%Y A185190 Cf.  A143136, A205886.
%K A185190 nonn,new
%O A185190 1,2
%A A185190 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A204579
%S A204579 1,1,1,4,5,1,36,49,14,1,576,820,273,30,1,14400,21076,7645,
%T A204579 1023,55,1,518400,773136,296296,44473,3003,91,1,25401600,
%U A204579 38402064,15291640,2475473,191620,7462,140,1,1625702400,2483133696
%V A204579 1,-1,1,4,-5,1,-36,49,-14,1,576,-820,273,-30,1,-14400,21076,-7645,
%W A204579 1023,-55,1,518400,-773136,296296,-44473,3003,-91,1,-25401600,
%X A204579 38402064,-15291640,2475473,-191620,7462,-140,1,1625702400,-2483133696
%N A204579 Triangle read by rows: matrix inverse of A036969.
%C A204579 This is a signed version of A008955 with rows in reverse order. - Peter Luschny, Feb 04 2012
%F A204579 T(n,k) = (-1)^(n-k)*A008955(n, n-k) - Peter Luschny, Feb 05 2012
%e A204579 Padding A036969 with zeros yields the infinite square matrix
%e A204579 [ 1  0  0  0 ...]
%e A204579 [ 1  1  0  0 ...]
%e A204579 [ 1  5  1  0 ...]
%e A204579 [ 1 21 14  0 ...]
%e A204579 with inverse
%e A204579 [  1  0  0  0 ...]
%e A204579 [ -1  1  0  0 ...]
%e A204579 [  4 -5  1  0 ...]
%e A204579 [-36 49 -14 1 ...].
%o A204579 (PARI) select(concat(Vec(matrix(10,10,n,k,T(n,k)/*from A036969*/)~^-1)), x->x)
%o A204579 (Sage)
%o A204579 def A204579(n, k) : return (-1)^(n-k)*A008955(n, n-k)
%o A204579     for n in (0..7) : print [A204579(n, k) for k in (0..n)] # Peter Luschny, Feb 05 2012
%Y A204579 Cf. A008955.
%K A204579 sign,tabl,new
%O A204579 1,4
%A A204579 M. F. Hasler (oeis2012-removeThis(AT)hasler.fr), Feb 03 2012
%E A204579 Typo in data corrected, Peter Luschny, Feb 05 2012

%I A206226
%S A206226 1,1,3,12,64,377,2432,16475,116263,845105,6292069,47759392,368379006,
%T A206226 2879998966,22777018771,181938716422,1465972415692,11902724768574,
%U A206226 97299665768397,800212617435074,6617003142869419,54985826573015541,458962108485797208,3846526994743330075
%N A206226 Number of partitions of n^2 into parts not greater than n.
%F A206226 a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).
%o A206226 (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
%o A206226 for(n=0,25,print1(a(n),", "))
%Y A206226 Cf. A173519, A206227, A206240, A107379.
%K A206226 nonn,new
%O A206226 0,3
%A A206226 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206227
%S A206227 1,1,4,19,108,674,4494,31275,225132,1662894,12541802,96225037,
%T A206227 748935563,5900502806,46976736513,377425326138,3056671009814,
%U A206227 24930725879856,204623068332997,1688980598900228,14012122025369431,116784468316023069,977437078888272796,8212186058546599006
%N A206227 Number of partitions of n^2+n into parts not greater than n.
%F A206227 a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).
%o A206227 (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2+n)))),n^2+n)}
%o A206227 for(n=0,30,print1(a(n),", "))
%Y A206227 Cf. A173519, A206226, A206240, A107379.
%K A206227 nonn,new
%O A206227 0,3
%A A206227 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206240
%S A206240 1,1,2,7,34,192,1206,8033,55974,403016,2977866,22464381,172388026,
%T A206240 1341929845,10573800028,84192383755,676491536028,5479185281572,
%U A206240 44692412971566,366844007355202,3028143252035976,25123376972033392,209401287806758273,1752674793617241002
%N A206240  Number of partitions of n^2-n into parts not greater than n.
%F A206240  a(n) = [x^(n^2-n)] Product_{k=1..n} 1/(1 - x^k).
%o A206240  (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2-n)))),n^2-n)}
%o A206240 for(n=0,30,print1(a(n),", "))
%Y A206240  Cf. A173519, A206226, A206227, A107379.
%K A206240 nonn,new
%O A206240 0,3
%A A206240 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206229
%S A206229 1,1,2,8,31,124,515,2166,9182,39195,168216,725043,3136223,13606891,
%T A206229 59187790,258034685,1127137141,4932071321,21614913239,94859273448,
%U A206229 416820578198,1833626307670,8074598332650,35591081565244,157013886785417,693237405812328
%N A206229 a(n) = [x^n] Product_{k=1..n} (1 + x^k)^(n-k+1).
%e A206229 Let [x^n] F(x) denote the coefficient of x^n in F(x); then
%e A206229 a(0) = 1;
%e A206229 a(1) = [x] (1+x) = 1;
%e A206229 a(2) = [x^2] (1+x)^2*(1+x^2) = 2;
%e A206229 a(3) = [x^3] (1+x)^3*(1+x^2)^2*(1+x^3) = 8;
%e A206229 a(4) = [x^4] (1+x)^4*(1+x^2)^3*(1+x^3)^2*(1+x^4) = 31; ...
%e A206229 as illustrated below.
%e A206229 The coefficients in Product_{k=1..n} (1+x^k)^(n-k+1) for n=0..9 begin:
%e A206229 n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
%e A206229 n=1: [1,(1), 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A206229 n=2: [1, 2,(2), 2, 1, 0, 0, 0, 0, 0, 0, 0 ...];
%e A206229 n=3: [1, 3, 5, (8), 10, 10, 10, 8, 5, 3, 1, 0 ...];
%e A206229 n=4: [1, 4, 9, 18, (31), 46, 64, 82, 96, 106, 110, 106 ...];
%e A206229 n=5: [1, 5, 14, 33, 68, (124), 210, 332, 492, 693, 931, ...];
%e A206229 n=6: [1, 6, 20, 54, 127, 266, (515), 934, 1597, 2602, ...];
%e A206229 n=7: [1, 7, 27, 82, 215, 502, 1078, (2166), 4109, 7428, ...];
%e A206229 n=8: [1, 8, 35, 118, 340, 870, 2038, 4454, (9182), 18020, ...];
%e A206229 n=9: [1, 9, 44, 163, 511, 1417, 3582, 8420, 18634,(39195), ...]; ...
%e A206229 where the coefficients in parenthesis start this sequence.
%o A206229 (PARI) {a(n)=polcoeff(prod(k=1,n,(1+x^k+x*O(x^n))^(n-k+1)),n)}
%o A206229 for(n=0,30,print1(a(n),", "))
%Y A206229 Cf. A206228.
%K A206229 nonn,new
%O A206229 0,3
%A A206229 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206228
%S A206228 1,1,4,17,80,384,1887,9385,47139,238488,1213588,6204547,31844710,
%T A206228 163978344,846741721,4382945317,22735196277,118151632006,615032941924,
%U A206228 3206257881171,16736910271178,87472908459696,457662760258109,2396899780970552,12564645719730297
%N A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).
%e A206228 Let [x^n] F(x) denote the coefficient of x^n in F(x); then
%e A206228 a(0) = 1;
%e A206228 a(1) = [x] 1/(1-x) = 1;
%e A206228 a(2) = [x^2] 1/((1-x)^2*(1-x^2)) = 4;
%e A206228 a(3) = [x^3] 1/((1-x)^3*(1-x^2)^2*(1-x^3)) = 17;
%e A206228 a(4) = [x^4] 1/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) = 80; ...
%e A206228 as illustrated below.
%e A206228 The coefficients in Product_{k=1..n} 1/(1-x^k)^(n-k+1) for n=0..9 begin:
%e A206228 n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
%e A206228 n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...];
%e A206228 n=2: [1, 2,(4), 6, 9, 12, 16, 20, 25, 30, 36, 42, ...];
%e A206228 n=3: [1, 3, 8, (17), 33, 58, 97, 153, 233, 342, 489, 681, ...];
%e A206228 n=4: [1, 4, 13, 34, (80), 170, 339, 636, 1141, 1964, 3270, ...];
%e A206228 n=5: [1, 5, 19, 58, 157,(384), 874, 1869, 3803, 7408, 13907, ...];
%e A206228 n=6: [1, 6, 26, 90, 273, 746, (1887), 4474, 10062, 21620, ...];
%e A206228 n=7: [1, 7, 34, 131, 438, 1314, 3632, (9385), 22940, 53466, ...];
%e A206228 n=8: [1, 8, 43, 182, 663, 2158, 6445, 17944, (47139), 117842, ...];
%e A206228 n=9: [1, 9, 53, 244, 960, 3361, 10757, 32008, 89651, (238488), ...]; ...
%e A206228 where the coefficients in parenthesis start this sequence.
%e A206228 Incidentally, the antidiagonal sums in the above table form A206119.
%o A206228 (PARI) {a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^n))^(n-k+1)),n)}
%o A206228 for(n=0,41,print1(a(n),", "))
%Y A206228 Cf. A206119, A206229.
%K A206228 nonn,new
%O A206228 0,3
%A A206228 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 05 2012

%I A206238
%S A206238 15,60,60,310,256,310,1640,1136,1136,1640,8910,5728,4456,5728,8910,
%T A206238 51066,31652,27168,27168,31652,51066,294546,170728,133392,283728,
%U A206238 133392,170728,294546,1710184,943584,607008,1236432,1236432,607008,943584,1710184
%N A206238 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206238 Table starts
%C A206238 ......15......60......310......1640........8910........51066........294546
%C A206238 ......60.....256.....1136......5728.......31652.......170728........943584
%C A206238 .....310....1136.....4456.....27168......133392.......607008.......3503136
%C A206238 ....1640....5728....27168....283728.....1236432......9042600......95322432
%C A206238 ....8910...31652...133392...1236432....10915392....118573968....1122086640
%C A206238 ...51066..170728...607008...9042600...118573968...1448239080...22535636736
%C A206238 ..294546..943584..3503136..95322432..1122086640..22535636736..649065145152
%C A206238 .1710184.5175034.17206032.419146392.10022726928.303011941944.8026428934128
%H A206238 R. H. Hardin, <a href="/A206238/b206238.txt">Table of n, a(n) for n = 1..544</a>
%F A206238 Empirical for column k:
%F A206238 k=1: a(n) = 8*a(n-1) -11*a(n-2) +36*a(n-3) -303*a(n-4) +232*a(n-5) +147*a(n-6) +756*a(n-7) for n>8
%F A206238 k=2: a(n) = 3*a(n-1) +20*a(n-2) -14*a(n-3) -133*a(n-4) +95*a(n-5) +123*a(n-6) +9*a(n-7) -102*a(n-8) for n>10
%F A206238 k=3: a(n) = a(n-1) +129*a(n-3) -129*a(n-4) for n>7
%F A206238 k=4: a(n) = a(n-1) +339*a(n-3) -339*a(n-4) for n>8
%F A206238 k=5: a(n) = a(n-1) +921*a(n-3) -921*a(n-4) for n>9
%F A206238 k=6: a(n) = a(n-1) +2571*a(n-3) -2571*a(n-4) for n>10
%F A206238 k=7: a(n) = a(n-1) +7329*a(n-3) -7329*a(n-4) for n>11
%F A206238 k=8: a(n) = a(n-1) +21219*a(n-3) -21219*a(n-4) for n>12
%F A206238 k=9: a(n) = a(n-1) +62121*a(n-3) -62121*a(n-4) for n>13
%F A206238 k=10: a(n) = a(n-1) +183291*a(n-3) -183291*a(n-4) for n>14
%F A206238 k=11: a(n) = a(n-1) +543729*a(n-3) -543729*a(n-4) for n>15
%F A206238 apparently a(n) = a(n-1) +3*A085279(k+1)*a(n-3) -3*A085279(k+1)*a(n-4) for k>2 and n>k+4
%e A206238 Some solutions for n=4 k=3
%e A206238 ..0..0..1..0....0..0..1..1....0..0..1..1....0..1..2..0....0..0..1..1
%e A206238 ..0..1..0..0....0..2..3..3....2..2..3..1....3..2..2..0....2..2..0..1
%e A206238 ..2..0..0..1....2..3..3..2....1..2..2..3....2..2..1..2....3..2..2..3
%e A206238 ..0..0..2..3....3..3..0..3....0..1..2..2....2..1..2..2....2..1..2..2
%e A206238 ..0..1..3..3....0..0..3..3....0..0..3..2....3..2..2..3....2..2..0..2
%K A206238 nonn,tabl,new
%O A206238 1,1
%A A206238 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206237
%S A206237 294546,943584,3503136,95322432,1122086640,22535636736,649065145152,
%T A206237 8026428934128,163621204516992,4735717646406528,58825697658136176,
%U A206237 1199179807904946432,34708074630513355776,431133538136479945968
%N A206237 Number of (n+1)X8 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206237 Column 7 of A206238
%H A206237 R. H. Hardin, <a href="/A206237/b206237.txt">Table of n, a(n) for n = 1..210</a>
%F A206237 Empirical: a(n) = a(n-1) +7329*a(n-3) -7329*a(n-4) for n>11
%e A206237 Some solutions for n=4
%e A206237 ..0..0..1..2..2..3..0..3....0..0..1..1..0..1..1..2....0..0..1..0..0..1..2..0
%e A206237 ..0..1..2..2..1..0..0..3....2..2..3..1..1..2..1..1....0..3..0..0..1..2..2..0
%e A206237 ..3..2..2..3..0..0..1..2....3..2..2..3..1..1..2..1....1..0..0..3..2..2..0..1
%e A206237 ..2..2..3..0..0..1..2..2....0..1..2..2..0..1..1..2....0..0..3..2..2..3..1..1
%e A206237 ..1..1..0..0..3..2..2..1....0..0..3..2..2..3..1..1....1..1..2..2..3..1..1..3
%K A206237 nonn,new
%O A206237 1,1
%A A206237 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206236
%S A206236 51066,170728,607008,9042600,118573968,1448239080,22535636736,
%T A206236 303011941944,3712941210144,57939122017416,779043702707184,
%U A206236 9545971851249384,148961482706745696,2002921359660139224
%N A206236 Number of (n+1)X7 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206236 Column 6 of A206238
%H A206236 R. H. Hardin, <a href="/A206236/b206236.txt">Table of n, a(n) for n = 1..210</a>
%F A206236 Empirical: a(n) = a(n-1) +2571*a(n-3) -2571*a(n-4) for n>10
%e A206236 Some solutions for n=4
%e A206236 ..0..0..1..0..0..1..0....0..0..1..0..0..1..0....0..0..1..0..0..1..2
%e A206236 ..0..2..0..0..2..0..0....1..0..0..1..0..0..1....0..1..0..0..3..2..2
%e A206236 ..1..0..0..2..0..0..2....2..3..0..0..1..0..0....2..0..0..3..2..2..1
%e A206236 ..0..0..1..0..0..3..1....2..2..1..0..0..1..1....0..0..1..2..2..3..0
%e A206236 ..3..3..0..0..2..1..1....3..2..2..3..0..1..2....3..3..2..2..3..0..0
%K A206236 nonn,new
%O A206236 1,1
%A A206236 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206235
%S A206235 8910,31652,133392,1236432,10915392,118573968,1122086640,10022726928,
%T A206235 109206613488,1033441784400,9230931489648,100579291011408,
%U A206235 951799883421360,8501687901954768,92633527021495728,876607692631061520
%N A206235 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206235 Column 5 of A206238
%H A206235 R. H. Hardin, <a href="/A206235/b206235.txt">Table of n, a(n) for n = 1..210</a>
%F A206235 Empirical: a(n) = a(n-1) +921*a(n-3) -921*a(n-4) for n>9
%e A206235 Some solutions for n=4
%e A206235 ..0..1..1..2..3..2....0..0..1..0..0..2....0..1..1..0..1..0....0..0..1..0..0..1
%e A206235 ..1..1..2..3..3..2....0..1..0..0..1..0....1..1..2..1..1..0....0..1..0..0..2..3
%e A206235 ..1..2..3..3..0..3....1..0..0..1..0..0....1..0..1..1..3..2....3..0..0..1..3..3
%e A206235 ..0..3..3..0..3..3....0..0..2..0..0..2....2..1..1..0..2..2....0..0..2..3..3..1
%e A206235 ..3..3..2..3..3..2....0..3..0..0..1..3....1..1..3..2..2..3....1..1..3..3..0..2
%K A206235 nonn,new
%O A206235 1,1
%A A206235 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206234
%S A206234 1640,5728,27168,283728,1236432,9042600,95322432,419146392,3065437344,
%T A206234 32314300392,142090622832,1039183255560,10954547828832,48168721135992,
%U A206234 352283123630784,3713591713969992,16329196465097232,119423978910831720
%N A206234 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206234 Column 4 of A206238
%H A206234 R. H. Hardin, <a href="/A206234/b206234.txt">Table of n, a(n) for n = 1..210</a>
%F A206234 Empirical: a(n) = a(n-1) +339*a(n-3) -339*a(n-4) for n>8
%e A206234 Some solutions for n=4
%e A206234 ..0..0..1..1..2....0..0..1..2..0....0..0..1..1..2....0..1..0..0..2
%e A206234 ..1..1..2..1..1....0..3..2..2..0....1..1..2..1..1....2..0..0..3..0
%e A206234 ..2..1..1..2..1....3..2..2..0..1....3..1..1..2..1....0..0..1..0..0
%e A206234 ..3..0..1..1..2....2..2..3..1..1....1..2..1..1..2....3..3..0..0..3
%e A206234 ..3..3..2..1..2....2..3..1..1..0....1..1..2..1..1....1..3..0..1..2
%K A206234 nonn,new
%O A206234 1,1
%A A206234 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206233
%S A206233 310,1136,4456,27168,133392,607008,3503136,17206032,78302496,
%T A206233 451903008,2219576592,10101020448,58295486496,286325378832,
%U A206233 1303031636256,7520117756448,36935973867792,168091081075488,970095190580256
%N A206233 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206233 Column 3 of A206238
%H A206233 R. H. Hardin, <a href="/A206233/b206233.txt">Table of n, a(n) for n = 1..210</a>
%F A206233 Empirical: a(n) = a(n-1) +129*a(n-3) -129*a(n-4) for n>7
%e A206233 Some solutions for n=4
%e A206233 ..0..1..2..0....0..0..1..0....0..0..1..0....0..0..1..1....0..0..1..0
%e A206233 ..3..2..2..0....0..1..0..0....2..0..0..2....1..1..0..1....2..0..0..1
%e A206233 ..2..2..1..2....1..0..0..1....3..1..0..0....2..1..1..0....3..1..0..0
%e A206233 ..2..1..2..2....0..0..2..3....3..3..1..0....1..2..1..1....3..3..1..0
%e A206233 ..3..2..2..3....1..1..3..3....1..3..3..1....1..1..0..0....0..3..3..1
%K A206233 nonn,new
%O A206233 1,1
%A A206233 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206232
%S A206232 60,256,1136,5728,31652,170728,943584,5175034,28475596,156707492,
%T A206232 861735284,4742758462,26087819020,143545024690,789722083016,
%U A206232 4344918701514,23904948840000,131519096655384,723594734816028
%N A206232 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206232 Column 2 of A206238
%H A206232 R. H. Hardin, <a href="/A206232/b206232.txt">Table of n, a(n) for n = 1..210</a>
%F A206232 Empirical: a(n) = 3*a(n-1) +20*a(n-2) -14*a(n-3) -133*a(n-4) +95*a(n-5) +123*a(n-6) +9*a(n-7) -102*a(n-8) for n>10
%e A206232 Some solutions for n=4
%e A206232 ..0..0..0....0..0..0....0..1..1....0..0..0....0..0..0....0..1..0....0..0..0
%e A206232 ..1..2..1....1..2..1....2..3..3....1..2..1....1..2..3....0..1..1....1..2..1
%e A206232 ..1..3..1....1..3..1....3..3..2....1..3..1....1..0..3....1..2..1....1..0..1
%e A206232 ..1..0..1....1..0..1....0..0..3....1..2..1....1..2..3....1..1..2....1..2..1
%e A206232 ..1..2..1....2..2..1....1..0..0....3..3..1....0..0..3....3..1..2....3..3..3
%K A206232 nonn,new
%O A206232 1,1
%A A206232 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206231
%S A206231 15,60,310,1640,8910,51066,294546,1710184,10051522,59273370,350336326,
%T A206231 2076929912,12328636710,73241168202,435453806538,2590088923960,
%U A206231 15409982499130,91703551575882,545793722630878,3248685323916392
%N A206231 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206231 Column 1 of A206238
%H A206231 R. H. Hardin, <a href="/A206231/b206231.txt">Table of n, a(n) for n = 1..210</a>
%F A206231 Empirical: a(n) = 8*a(n-1) -11*a(n-2) +36*a(n-3) -303*a(n-4) +232*a(n-5) +147*a(n-6) +756*a(n-7) for n>8
%e A206231 Some solutions for n=4
%e A206231 ..0..1....0..0....0..1....0..0....0..0....0..1....0..1....0..0....0..0....0..0
%e A206231 ..0..0....0..1....0..0....0..1....1..0....2..1....2..0....1..1....0..1....0..1
%e A206231 ..0..1....1..2....0..1....1..0....0..1....3..1....0..0....2..1....1..2....1..2
%e A206231 ..1..2....1..1....1..0....0..0....0..0....0..1....3..0....1..0....2..2....1..1
%e A206231 ..1..1....0..1....0..0....2..2....0..2....3..3....2..1....0..0....1..1....0..0
%K A206231 nonn,new
%O A206231 1,1
%A A206231 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206230
%S A206230 15,256,4456,283728,10915392,1448239080,649065145152,203396423662488,
%T A206230 222950396746461504,842104767130879914696,2305567526828592241306992,
%U A206230 22136848571138455592306188104,737206270850592131066195723493312
%N A206230 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..3 introduced in row major order
%C A206230 Diagonal of A206238
%H A206230 R. H. Hardin, <a href="/A206230/b206230.txt">Table of n, a(n) for n = 1..16</a>
%e A206230 Some solutions for n=4
%e A206230 ..0..1..0..0..1....0..1..0..0..1....0..0..1..2..0....0..1..2..1..0
%e A206230 ..2..0..0..1..0....0..0..2..0..0....0..3..2..2..0....1..2..2..1..1
%e A206230 ..0..0..1..0..0....1..0..0..3..0....3..2..2..0..1....2..2..0..2..2
%e A206230 ..2..2..0..0..1....2..3..0..0..1....2..2..3..1..1....0..0..2..2..3
%e A206230 ..3..2..0..2..0....2..2..3..0..0....2..3..1..1..0....3..0..2..3..0
%K A206230 nonn,new
%O A206230 1,1
%A A206230 R. H. Hardin (rhhardin(AT)att.net) Feb 05 2012

%I A206156
%S A206156 1,2,6,92,5410,1400652,2687407464,18947436116184,536104663173431874,
%T A206156 130559883231879141946580,136031455187223511721647272376,
%U A206156 483565526783420050082035900177878504,14487924180895151383693101563813954330590756
%N A206156 a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
%C A206156 Ignoring initial term a(0), equals the logarithmic derivative of A206155.
%e A206156 L.g.f.: L(x) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
%e A206156 where exponentiation yields A206155:
%e A206156 exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
%e A206156 Illustration of initial terms:
%e A206156 a(1) = 1^0 + 1^2 = 2;
%e A206156 a(2) = 1^0 + 2^2 + 1^4 = 6;
%e A206156 a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
%e A206156 a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
%e A206156 a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...
%o A206156 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(2*k))}
%o A206156 for(n=0,16,print1(a(n),", "))
%Y A206156 Cf. A206155 (exp), A184731, A206154, A206158, A206152.
%K A206156 nonn,new
%O A206156 0,2
%A A206156 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206155
%S A206155 1,2,5,38,1425,283002,448468978,2707673843860,67018498701021670,
%T A206155 14506787732148113566364,13603174532364904984495776225,
%U A206155 43960529641219941452921634596223366,1207327102995668834632770987833295579308107,188859837731175560954429490131760211759694331013582
%N A206155 G.f.: exp( Sum_{n>=1} A206156(n)*x^n/n ), where A206156(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
%C A206155 Logarithmic derivative yields A206156.
%e A206155 G.f.: A(x) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 +...
%e A206155 where the logarithm of the g.f. begins:
%e A206155 log(A(x)) = 2*x + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 + 2687407464*x^6/6 +...+ A206156(n)*x^n/n +...
%o A206155 (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k-0))+x*O(x^n))),n)}
%o A206155 for(n=0,16,print1(a(n),", "))
%Y A206155 Cf. A206156 (log), A184730, A206153, A206157, A206151.
%K A206155 nonn,new
%O A206155 0,2
%A A206155 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206158
%S A206158 1,2,10,272,24226,12053252,40086916024,429254371605824,
%T A206158 23527609330364490754,10714627376371224032350052,
%U A206158 16964729291782419425708732425300,109783535843179466164398767001178968704,6782057095273243388704415924996348722446049600
%N A206158  a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).
%C A206158  Ignoring initial term a(0), equals the logarithmic derivative of A206157.
%e A206158  L.g.f.: L(x) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 +...
%e A206158 where exponentiation yields A206157:
%e A206158 exp(L(x)) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 + 6686021554*x^6 +...
%e A206158 Illustration of initial terms:
%e A206158 a(1) = 1^1 + 1^3 = 2;
%e A206158 a(2) = 1^1 + 2^3 + 1^5 = 10;
%e A206158 a(3) = 1^1 + 3^3 + 3^5 + 1^7 = 272;
%e A206158 a(4) = 1^1 + 4^3 + 6^5 + 4^7 + 1^9 = 24226;
%e A206158 a(5) = 1^1 + 5^3 + 10^5 + 10^7 + 5^9 + 1^11 = 12053252; ...
%o A206158  (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(2*k+1))}
%o A206158 for(n=0,16,print1(a(n),", "))
%Y A206158  Cf. A206157 (exp), A184731, A206154, A206156, A206152.
%K A206158 nonn,new
%O A206158 0,2
%A A206158 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206157
%S A206157 1,2,7,102,6261,2423430,6686021554,61335432894584,2941073857435300366,
%T A206157 1190520035262419577871332,1696475310227140760623646031573,
%U A206157 9980324833243234634513255755001535870,565171444566758371735408026461987217216896790
%N A206157  G.f.: exp( Sum_{n>=1} A206158(n)*x^n/n ), where A206158(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).
%C A206157  Logarithmic derivative yields A206158.
%e A206157  G.f.: A(x) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 +...
%e A206157 where the logarithm of the g.f. begins:
%e A206157 log(A(x)) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 + 40086916024*x^6/6 +...+ A206158(n)*x^n/n +...
%o A206157  (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(2*k+1))+x*O(x^n))),n)}
%o A206157 for(n=0,16,print1(a(n),", "))
%Y A206157  Cf. A206158 (log), A184730, A206153, A206155, A206151.
%K A206157 nonn,new
%O A206157 0,2
%A A206157 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206154
%S A206154 1,2,10,110,2386,125752,14921404,3697835668,2223231412546,
%T A206154 3088517564289836,9040739066816429380,63462297965044771663708,
%U A206154 1064766030857977088480630740,37863276208844960432962611293828,3144384748384240804260912067907833280
%N A206154  a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
%C A206154  Ignoring initial term a(0), equals the logarithmic derivative of A206153.
%e A206154  L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
%e A206154 where exponentiation yields A206151:
%e A206154 exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
%e A206154 Illustration of initial terms:
%e A206154 a(1) = 1^2 + 1^3 = 2;
%e A206154 a(2) = 1^2 + 2^3 + 1^4 = 10;
%e A206154 a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
%e A206154 a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
%e A206154 a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...
%o A206154  (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
%o A206154 for(n=0,16,print1(a(n),", "))
%Y A206154  Cf. A206153 (exp), A184731, A206156, A206158, A206152.
%K A206154 nonn,new
%O A206154 0,2
%A A206154 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206153
%S A206153 1,2,7,48,693,26632,2542514,533442978,278979307990,343728261289376,
%T A206153 904762216681139381,5771110378770242683658,88742047516327429085056353,
%U A206153 2912737209806573079629325613400,224604736339682169442980060945290802
%N A206153  G.f.: exp( Sum_{n>=1} A206154(n)*x^n/n ), where A206154(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
%C A206153  Logarithmic derivative yields A206154.
%e A206153  G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
%e A206153 where the logarithm of the g.f. begins:
%e A206153 log(A(x)) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 + 14921404*x^6/6 +...+ A206154(n)*x^n/n +...
%o A206153  (PARI) {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=0,m,binomial(m,k)^(k+2))+x*O(x^n))),n)}
%o A206153 for(n=0,16,print1(a(n),", "))
%Y A206153  Cf. A206154 (log), A184730, A206155, A206157, A206151.
%K A206153 nonn,new
%O A206153 0,2
%A A206153 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A205578
%S A205578 0,0,0,0,0,0,0,0,1,9,221
%N A205578 Number of n-node simple graphs having clique number 9
%K A205578 nonn,new
%O A205578 1,10
%A A205578 Michael Sollami (michaelsollami(AT)gmail.com), Jan 29 2012

%I A205567
%S A205567 0,0,0,0,0,0,0,1,8,165,8214
%N A205567 Number of 8-chromatic (i.e. chromatic number equals 8) simple graphs on n nodes.
%H A205567 Keith M. Briggs, <a href="http://keithbriggs.info/cgt.html">Combinatorial Graph Theory</a>
%H A205567 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/n-ChromaticGraph.html">n-Chromatic Graph</a>
%Y A205567 Cf. A076278, A076279, A076280, A076281.
%K A205567 nonn,more,new
%O A205567 1,9
%A A205567 Michael Sollami (michaelsollami(AT)gmail.com), Jan 28 2012

%I A205976
%S A205976 1,4,12,32,84,120,384,52,1260,1768,3960,4272,16128,13048,4524,58560,
%T A205976 122388,114984,403104,334480,1136520,175136,2550384,2751072,11128320,
%U A205976 9303100,20394024,31426880,8898708,61707480,239627520,172322432,548933868,676718976,1231823592
%N A205976  a(n) = Fibonacci(n)*A028594(n) for n>=1, with a(0)=1, where A028594 lists the coefficients in  (theta_3(x)*theta_3(7*x)+theta_2(x)*theta_2(7*x))^2.
%C A205976  Compare g.f. to the Lambert series of A028594:
%C A205976 1 + 4*Sum_{n>=1} Chi(n,7)*n*x^n/(1-x^n).
%C A205976 Here Chi(n,7) = principal Dirichlet character of n modulo 7.
%F A205976  G.f.: 1 + 4*Sum_{n>=1} Fibonacci(n)*Chi(n,7)*n*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205976  G.f.: A(x) = 1 + 4*x + 12*x^2 + 32*x^3 + 84*x^4 + 120*x^5 + 384*x^6 + 52*x^7 +...
%e A205976 where A(x) = 1 + 1*4*x + 1*12*x^2 + 2*16*x^3 + 3*28*x^4 + 5*24*x^5 + 8*48*x^6 + 13*4*x^7 + 21*60*x^8 + 34*52*x^9 +...+ Fibonacci(n)*A028594(n)*x^n +...
%e A205976 The g.f. is also given by the identity:
%e A205976 A(x) = 1 + 4*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1-3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1-7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1-18*x^6+x^12) + 0*13*7*x^7/(1+29*x^7-x^14) +...).
%e A205976 The values of the Dirichlet character Chi(n,7) repeat [1,1,1,1,1,1,0, ...].
%o A205976  (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205976 {a(n)=polcoeff(1 + 4*sum(m=1,n,fibonacci(m)*kronecker(m,7)^2*m*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205976 for(n=0,60,print1(a(n),", "))
%Y A205976  Cf. A028594, A205975, A203847, A000204 (Lucas).
%K A205976 nonn,new
%O A205976 0,2
%A A205976 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A205975
%S A205975 1,2,4,0,18,0,0,26,168,68,0,356,0,0,1508,0,9870,0,10336,0,0,0,141688,
%T A205975 114628,0,150050,0,0,1906866,2056916,0,0,26139708,0,0,0,89582112,
%U A205975 96631268,0,0,0,0,0,1733977748,8416904796,0,14690495224,0,0,15557484098
%N A205975 a(n) = Fibonacci(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].
%C A205975 Compare the g.f. to the Lambert series of A002652:
%C A205975 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).
%F A205975 G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205975 G.f.: A(x) = 1 + 2*x + 4*x^2 + 18*x^4 + 26*x^7 + 168*x^8 + 68*x^9 + 356*x^11 +...
%e A205975 where A(x) = 1 + 1*2*x + 1*4*x^2 + 3*6*x^4 + 14*2*x^7 + 21*8*x^8 + 34*2*x^9 + 89*4*x^11 + 377*4*x^14 + 987*10*x^16 +...+ Fibonacci(n)*A002652(n)*x^n +...
%e A205975 The g.f. is also given by the identity:
%e A205975 A(x) = 1 + 2*( 1*x/(1-x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1-11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
%e A205975 The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
%o A205975 (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205975 {a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205975 for(n=0,60,print1(a(n),", "))
%Y A205975 Cf. A002652, A205974, A205976, A203847, A000204 (Lucas).
%K A205975 nonn,new
%O A205975 0,2
%A A205975 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A205974
%S A205974 1,2,0,0,6,0,0,26,84,68,0,356,0,0,0,0,5922,0,0,0,0,0,0,114628,0,
%T A205974 150050,0,0,635622,2056916,0,0,17426472,0,0,0,29860704,96631268,0,0,0,
%U A205974 0,0,1733977748,2805634932,0,0,0,0,15557484098,0,0,0,213265164692,0,0
%N A205974  a(n) = Fibonacci(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).
%C A205974  Compare g.f. to the Lambert series of A033719:
%C A205974 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n).
%F A205974  G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*(-x)^n + (-1)^n*x^(2*n)).
%e A205974  G.f.: A(x) = 1 + 2*x + 6*x^4 + 26*x^7 + 84*x^8 + 68*x^9 + 356*x^11 +...
%e A205974 where A(x) = 1 + 1*2*x + 3*2*x^4 + 13*2*x^7 + 21*4*x^8 + 34*2*x^9 + 89*4*x^11 + 987*6*x^16 + 28657*4*x^23 +...+ Fibonacci(n)*A033719(n)*x^n +...
%e A205974 The g.f. is also given by the identity:
%e A205974 A(x) = 1 + 2*( 1*x/(1+x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1+4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
%e A205974 The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].
%o A205974  (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205974 {a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205974 for(n=0,40,print1(a(n),", "))
%Y A205974  Cf. A033719, A205973, A205975, A203847, A000204 (Lucas).
%K A205974 nonn,new
%O A205974 0,2
%A A205974 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A205973
%S A205973 1,9,27,18,351,1080,216,5850,9639,306,35640,96120,16848,
%T A205973 356490,508950,131760,1821015,4139424,69768,13621698,18996120,
%U A205973 4925700,57383640,136178064,21282912,405810225,557193870,1767762,1859194350,3887571240,539161920
%V A205973 1,-9,27,-18,-351,1080,216,-5850,9639,-306,-35640,96120,-16848,
%W A205973 -356490,508950,131760,-1821015,4139424,69768,-13621698,18996120,
%X A205973 -4925700,-57383640,136178064,21282912,-405810225,557193870,-1767762,-1859194350,3887571240,-539161920
%N A205973  a(n) = Fibonacci(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.
%C A205973  Compare the g.f. to the Lambert series of A109041:
%C A205973 1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).
%F A205973  G.f.: 1 - 9*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*n^2*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205973  G.f.: A(x) = 1 - 9*x + 27*x^2 - 18*x^3 - 351*x^4 + 1080*x^5 + 216*x^6 +...
%e A205973 where A(x) = 1 - 1*9*x + 1*27*x^2 - 2*9*x^3 - 3*117*x^4 + 5*216*x^5 + 8*27*x^6 - 13*450*x^7 + 21*459*x^8 +...+ Fibonacci(n)*A109041(n)*^n +...
%e A205973 The g.f. is also given by the identity:
%e A205973 A(x) = 1 - 9*( 1*1*x/(1-x-x^2) - 1*4*x^2/(1-3*x^2+x^4) + 3*16*x^4/(1-7*x^4+x^8) - 5*25*x^5/(1-11*x^5-x^10) + 13*49*x^7/(1-29*x^7-x^14) - 21*64*x^8/(1-47*x^8+x^16) +...).
%e A205973 The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%o A205973  (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205973 {a(n)=polcoeff(1 - 9*sum(m=1,n,fibonacci(m)*kronecker(m,3)*m^2*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205973 for(n=0,40,print1(a(n),", "))
%Y A205973  Cf. A109041, A205972, A205974, A203847, A000204 (Lucas).
%K A205973 sign,new
%O A205973 0,2
%A A205973 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A205972
%S A205972 1,6,12,12,18,0,96,156,252,204,0,0,864,2796,9048,0,5922,0,
%T A205972 31008,50172,0,131352,0,0,556416,450150,2913432,1178508,3813732,
%U A205972 0,0,16155228,26139708,0,0,0,89582112,289893804,938116056,758951832,0,0,6429943104
%V A205972 1,-6,12,-12,-18,0,96,-156,252,-204,0,0,-864,-2796,9048,0,-5922,0,
%W A205972 31008,-50172,0,-131352,0,0,556416,-450150,2913432,-1178508,-3813732,
%X A205972 0,0,-16155228,26139708,0,0,0,-89582112,-289893804,938116056,-758951832,0,0,6429943104
%N A205972  a(n) = Fibonacci(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.
%C A205972  Compare the g.f. to the Lambert series of A122859:
%C A205972 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).
%F A205972  G.f.: 1 - 6*Sum_{n>=1} Fibonacci(n)*Kronecker(n,3)*x^n/(1 + Lucas(n)*x^n + (-1)^n*x^(2*n)).
%e A205972  G.f.: A(x) = 1 - 6*x + 12*x^2 - 12*x^3 - 18*x^4 + 96*x^6 - 156*x^7 +...
%e A205972 where A(x) = 1 - 1*6*x + 1*12*x^2 - 2*6*x^3 - 3*6*x^4 + 8*12*x^6 - 13*12*x^7 + 21*12*x^8 - 34*6*x^9 +...+ Fibonacci(n)*A122859(n)*x^n +...
%e A205972 The g.f. is also given by the identity:
%e A205972 A(x) = 1 - 6*( 1*x/(1+x-x^2) - 1*x^2/(1+3*x^2+x^4) + 3*x^4/(1+7*x^4+x^8) - 5*x^5/(1+11*x^5-x^10) + 13*x^7/(1+29*x^7-x^14) - 21*x^8/(1+47*x^8+x^16) +...).
%e A205972 The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].
%o A205972  (PARI) {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A205972 {a(n)=polcoeff(1 - 6*sum(m=1,n,fibonacci(m)*kronecker(m,3)*x^m/(1+Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
%o A205972 for(n=0,40,print1(a(n),", "))
%Y A205972  Cf. A122859, A205971, A205973, A205966, A205969, A203847, A000204 (Lucas).
%K A205972 sign,new
%O A205972 0,2
%A A205972 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206178
%S A206178 1,3,21,171,1521,14283,138909,1385163,14072193,145039923,1512191781,
%T A206178 15914734443,168802010001,1802247516891,19350710547021,
%U A206178 208783189719531,2262263134211073,24604815145831011,268499713118585781,2938736789722114731,32250788066104022961
%N A206178 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 2^k.
%C A206178 Ignoring initial term, equals the logarithmic derivative of A206177.
%C A206178 Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
%F A206178  a(2*3^n) == 3 (mod 9) for n>=0; a(n) == 0 (mod 9) if n/2 > 1 is not a power of 3.
%e A206178 L.g.f.: L(x) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 +...
%e A206178 Exponentiation equals the g.f. of A206177:
%e A206178 exp(L(x)) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...
%o A206178 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^3*2^k)}
%Y A206178 Cf. A206177, A000172, A206180.
%K A206178 nonn,new
%O A206178 0,2
%A A206178 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206180
%S A206180 1,4,34,352,3946,46744,573616,7217536,92527738,1203467464,15834369244,
%T A206180 210304283776,2815055712496,37930536447808,513972867056704,
%U A206180 6998587355233792,95704396144575898,1313665229153722408,18091969874675059204,249908773119244105792
%N A206180  a(n) = Sum_{k=0..n} binomial(n,k)^3 * 3^k.
%C A206180  Ignoring initial term, equals the logarithmic derivative of A206179.
%C A206180 Compare to Franel numbers: A000172(n) = Sum_{k=0..n} binomial(n,k)^3.
%e A206180  L.g.f.: L(x) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 +...
%e A206180 Exponentiation equals the g.f. of A206179:
%e A206180 exp(L(x)) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 + 175936*x^6 +...
%o A206180  (PARI) {a(n)=sum(k=0,n,binomial(n,k)^3*3^k)}
%o A206180 for(n=0,41,print1(a(n),", "))
%Y A206180  Cf. A206179, A000172, A206178.
%K A206180 nonn,new
%O A206180 0,2
%A A206180 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206179
%S A206179 1,4,25,196,1747,16996,175936,1907224,21423385,247515796,2925668236,
%T A206179 35239199704,431207470105,5347823877172,67093724913313,
%U A206179 850241358959044,10869754843088962,140045452765874704,1816842996684686656,23716478425653945472,311314468637807994391
%N A206179  G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^3 * 3^k ).
%C A206179  Logarithmic derivative yields A206180.
%e A206179  G.f.: A(x) = 1 + 4*x + 25*x^2 + 196*x^3 + 1747*x^4 + 16996*x^5 +...
%e A206179 where
%e A206179 log(A(x)) = 4*x + 34*x^2/2 + 352*x^3/3 + 3946*x^4/4 + 46744*x^5/5 + 573616*x^6/6 +...+ A206180(n)*x^n/n +...
%o A206179  (PARI) {a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^3*3^k))+x*O(x^n)),n)}
%o A206179 for(n=0,41,print1(a(n),", "))
%Y A206179  Cf. A206180, A206177.
%K A206179 nonn,new
%O A206179 0,2
%A A206179 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A206177
%S A206177 1,3,15,93,657,5067,41579,357297,3181305,29133387,272939679,
%T A206177 2605588317,25269158105,248367451299,2469462766347,24800305889217,
%U A206177 251258730935697,2565372042688563,26373806952805519,272818956588097341,2837840577104379201,29667671262881320347
%N A206177  G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^3 * 2^k ).
%C A206177  Logarithmic derivative yields A206178.
%e A206177  G.f.: A(x) = 1 + 3*x + 15*x^2 + 93*x^3 + 657*x^4 + 5067*x^5 + 41579*x^6 +...
%e A206177 where
%e A206177 log(A(x)) = 3*x + 21*x^2/2 + 171*x^3/3 + 1521*x^4/4 + 14283*x^5/5 + 138909*x^6/6 +...+ A206178(n)*x^n/n +...
%o A206177  (PARI) {a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m,k)^3*2^k))+x*O(x^n)),n)}
%Y A206177  Cf. A206178.
%K A206177 nonn,new
%O A206177 0,2
%A A206177 Paul D. Hanna (pauldhanna(AT)juno.com), Feb 04 2012

%I A195823
%S A195823 5,6,1,9,8,5,1,7,8,4,8,3,2,5,8,1,1,1,4,5,2,5,0,9,9,7,1,4,5,6,3,9,1,5,
%T A195823 8,3,9,5,7,3,2,0,7,3,4,9,6,5,3,7,6,1,9,3,5,9,9,1,7,7,6,8,2,0,9,5,3,7,
%U A195823 1,3,5,0,4,4,9,5,3,5,0,5,6,4,0,8,3,3,1
%N A195823 Decimal expansion of 8*pi*5^(1/2).
%H A195823 Soon-Yi Kang and Chang Heon Kim, <a href="http://arxiv.org/abs/0904.3777">Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups</a>, arXiv:0904.3777v1, Apr 24 2009, p. 5 (Example 1).
%e A195823 56.1985178483258111452509971456391583957320734965376193599177682...
%t A195823 RealDigits[8*Pi*5^(1/2), 10, 87][[1]] (* Bruno Berselli, Feb 04 2012 *)
%Y A195823 Cf. A000796.
%K A195823 nonn,cons,new
%O A195823 2,1
%A A195823 Omar E. Pol (info(AT)polprimos.com), Feb 01 2012
%E A195823 More terms from Bruno Berselli, Feb 04 2012

%I A206223
%S A206223 81,452,452,1970,7313,1970,8128,73133,73133,8128,33386,656053,1782191,
%T A206223 656053,33386,139390,5862927,35625711,35625711,5862927,139390,588254,
%U A206223 53835891,600064928,1827047303,600064928,53835891,588254,2496440
%N A206223 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206223 Table starts
%C A206223 ......81........452..........1970.............8128...............33386
%C A206223 .....452.......7313.........73133...........656053.............5862927
%C A206223 ....1970......73133.......1782191.........35625711...........600064928
%C A206223 ....8128.....656053......35625711.......1827047303.........74479700228
%C A206223 ...33386....5862927.....600064928......74479700228.......7626209039519
%C A206223 ..139390...53835891....9098547899....2537225006229.....656654900321262
%C A206223 ..588254..499752207..131428029860...77008927829374...46954627739500046
%C A206223 .2496440.4654712482.1883474084204.2231343350258712.2857994610813268269
%H A206223 R. H. Hardin, <a href="/A206223/b206223.txt">Table of n, a(n) for n = 1..84</a>
%e A206223 Some solutions for n=4 k=3
%e A206223 ..0..1..2..0....0..0..2..1....2..2..0..1....1..2..2..0....2..0..0..1
%e A206223 ..2..1..0..0....2..0..2..1....1..0..0..1....0..0..0..0....2..0..0..1
%e A206223 ..2..0..0..1....1..0..1..1....1..0..0..0....2..2..2..0....2..0..0..0
%e A206223 ..0..0..0..0....2..0..1..2....1..2..1..0....0..0..0..0....0..0..0..1
%e A206223 ..1..1..2..1....1..0..0..1....0..2..2..2....1..0..2..0....0..0..0..2
%Y A206223 Column 1 is A205208
%K A206223 nonn,tabl,new
%O A206223 1,1
%A A206223 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206222
%S A206222 588254,499752207,131428029860,77008927829374,46954627739500046,
%T A206222 33632275642926148580
%N A206222 Number of (n+1)X8 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206222 Column 7 of A206223
%e A206222 Some solutions for n=4
%e A206222 ..0..2..1..2..2..0..0..0....2..0..2..1..0..1..1..2....1..2..0..0..0..0..2..1
%e A206222 ..2..2..0..0..0..0..1..0....2..0..0..0..0..1..1..2....1..1..0..2..0..0..1..1
%e A206222 ..0..0..0..2..1..0..2..2....2..0..1..0..0..1..1..2....1..0..0..2..0..0..1..2
%e A206222 ..2..1..0..2..1..0..1..2....0..0..2..0..0..0..0..0....0..0..0..2..0..0..2..2
%e A206222 ..1..2..0..0..0..0..2..0....0..0..0..0..2..1..1..0....2..2..0..2..0..0..2..2
%K A206222 nonn,new
%O A206222 1,1
%A A206222 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206221
%S A206221 139390,53835891,9098547899,2537225006229,656654900321262,
%T A206221 162241662044855141,33632275642926148580,5657294413089823562871
%N A206221 Number of (n+1)X7 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206221 Column 6 of A206223
%e A206221 Some solutions for n=4
%e A206221 ..2..0..0..1..2..2..2....2..2..0..0..1..2..1....2..0..0..2..2..2..1
%e A206221 ..1..0..0..0..0..0..0....2..0..0..0..1..2..1....1..0..0..2..2..2..1
%e A206221 ..1..0..1..2..0..2..1....1..0..2..0..1..2..1....2..0..0..1..1..2..2
%e A206221 ..1..0..1..2..0..0..0....1..0..1..0..1..2..1....1..0..0..1..2..0..2
%e A206221 ..0..0..1..2..0..2..0....1..2..2..0..1..2..1....2..0..0..0..1..2..2
%K A206221 nonn,new
%O A206221 1,1
%A A206221 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206220
%S A206220 33386,5862927,600064928,74479700228,7626209039519,656654900321262,
%T A206220 46954627739500046,2857994610813268269,153669461683842088404,
%U A206220 7633859374178156693439,365363140359338927395032
%N A206220 Number of (n+1)X6 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206220 Column 5 of A206223
%H A206220 R. H. Hardin, <a href="/A206220/b206220.txt">Table of n, a(n) for n = 1..124</a>
%e A206220 Some solutions for n=4
%e A206220 ..1..2..1..0..0..2....0..1..0..2..1..0....0..1..2..1..0..0....2..0..0..1..0..0
%e A206220 ..1..2..1..0..0..1....0..0..0..0..0..0....1..1..2..1..0..0....1..0..0..1..0..2
%e A206220 ..0..0..0..0..0..1....0..1..0..1..2..2....0..0..0..0..0..0....0..0..0..0..0..1
%e A206220 ..0..1..1..2..0..0....0..0..0..0..0..0....0..1..2..1..0..0....0..1..0..0..0..1
%e A206220 ..0..0..0..0..0..0....2..2..1..0..2..2....0..0..0..0..0..0....0..0..0..1..0..2
%K A206220 nonn,new
%O A206220 1,1
%A A206220 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206219
%S A206219 8128,656053,35625711,1827047303,74479700228,2537225006229,
%T A206219 77008927829374,2231343350258712,64389782469709783,
%U A206219 1877157827258522194,55170332039640610650,1627763387646380404023,48095430422740484129183
%N A206219 Number of (n+1)X5 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206219 Column 4 of A206223
%H A206219 R. H. Hardin, <a href="/A206219/b206219.txt">Table of n, a(n) for n = 1..210</a>
%F A206219 Empirical: a(n) = 103*a(n-1) -4219*a(n-2) +81147*a(n-3) -397332*a(n-4) -14503310*a(n-5) +329978018*a(n-6) -2633737466*a(n-7) -4621703737*a(n-8) +283589064843*a(n-9) -2411136838927*a(n-10) +4991691652703*a(n-11) +67278935952078*a(n-12) -620599579539428*a(n-13) +1749152997998376*a(n-14) +5269824026917280*a(n-15) -59102318926312640*a(n-16) +172656732006803840*a(n-17) +92541855223544832*a(n-18) -2145453207148177664*a(n-19) +5848239068991970816*a(n-20) -854981061398701056*a(n-21) -32533098292024866816*a(n-22) +77324864401822253056*a(n-23) -26377215092523335680*a(n-24) -203801804660199915520*a(n-25) +404090179964000272384*a(n-26) -172703174480063627264*a(n-27) -423971470763460919296*a(n-28) +707497817831859290112*a(n-29) -362831080049269014528*a(n-30) -75679024184612093952*a(n-31) +150854984897157660672*a(n-32) -44931349155019751424*a(n-33) for n>51
%e A206219 Some solutions for n=4
%e A206219 ..0..0..2..2..1....2..2..0..0..1....2..1..1..1..2....1..1..0..2..0
%e A206219 ..0..0..0..0..0....1..0..0..0..2....2..0..0..0..1....0..0..0..0..0
%e A206219 ..1..2..1..0..2....1..0..1..0..2....1..0..0..2..0....0..2..1..2..0
%e A206219 ..0..0..0..0..2....1..2..1..0..2....2..0..0..1..2....0..0..0..0..0
%e A206219 ..1..2..1..0..2....0..2..2..0..0....0..2..0..1..2....0..0..1..1..0
%K A206219 nonn,new
%O A206219 1,1
%A A206219 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012 2012

%I A206218
%S A206218 1970,73133,1782191,35625711,600064928,9098547899,131428029860,
%T A206218 1883474084204,27236695360128,398320420587108,5870939348495954,
%U A206218 86941313422965979,1290991288648976502,19200795902731377026
%N A206218 Number of (n+1)X4 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206218 Column 3 of A206223
%H A206218 R. H. Hardin, <a href="/A206218/b206218.txt">Table of n, a(n) for n = 1..210</a>
%F A206218 Empirical: a(n) = 42*a(n-1) -625*a(n-2) +2990*a(n-3) +19429*a(n-4) -300482*a(n-5) +1205989*a(n-6) +739674*a(n-7) -21921768*a(n-8) +68928624*a(n-9) -44135280*a(n-10) -218575584*a(n-11) +570862080*a(n-12) -547928064*a(n-13) +191102976*a(n-14) for n>30
%e A206218 Some solutions for n=4
%e A206218 ..0..1..0..1....2..0..0..1....2..2..0..1....1..2..1..2....0..0..2..1
%e A206218 ..2..0..0..0....2..0..0..1....1..0..0..1....1..1..0..2....2..0..2..1
%e A206218 ..1..0..1..1....2..0..0..0....1..0..0..0....1..1..2..2....1..0..1..1
%e A206218 ..1..0..2..2....0..0..0..1....1..2..1..0....1..2..1..1....2..0..1..2
%e A206218 ..2..2..0..0....0..0..0..2....0..2..2..2....0..2..1..1....1..0..0..1
%K A206218 nonn,new
%O A206218 1,1
%A A206218 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206217
%S A206217 452,7313,73133,656053,5862927,53835891,499752207,4654712482,
%T A206217 43385661909,404522337676,3772143906359,35177251141094,
%U A206217 328055398640436,3059417909281589,28532069184985757,266090406288547230
%N A206217 Number of (n+1)X3 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206217 Column 2 of A206223
%H A206217 R. H. Hardin, <a href="/A206217/b206217.txt">Table of n, a(n) for n = 1..210</a>
%F A206217 Empirical: a(n) = 15*a(n-1) -49*a(n-2) -97*a(n-3) +664*a(n-4) -980*a(n-5) +448*a(n-6) for n>16
%e A206217 Some solutions for n=4
%e A206217 ..2..1..2....0..0..0....1..2..1....1..1..2....0..2..2....1..2..1....0..1..2
%e A206217 ..2..0..0....0..0..0....0..0..0....0..0..0....1..2..1....2..2..1....1..0..0
%e A206217 ..0..0..1....0..2..2....1..1..0....1..2..0....0..0..0....2..1..2....1..0..2
%e A206217 ..1..0..0....0..1..1....0..0..0....1..2..2....0..2..0....2..2..0....0..0..1
%e A206217 ..0..1..2....0..1..2....1..1..1....0..2..2....0..1..1....0..2..1....2..0..2
%K A206217 nonn,new
%O A206217 1,1
%A A206217 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206216
%S A206216 81,7313,1782191,1827047303,7626209039519,162241662044855141
%N A206216 Number of (n+1)X(n+1) 0..2 arrays with rows and columns of determinants of all 2X2 subblocks lexicographically nondecreasing
%C A206216 Diagonal of A206223
%e A206216 Some solutions for n=4
%e A206216 ..1..1..0..2..0....2..2..1..1..2....2..2..1..2..0....1..0..0..2..2
%e A206216 ..0..0..0..0..0....0..0..0..0..0....2..1..0..0..0....0..0..0..0..0
%e A206216 ..0..2..1..2..0....1..0..0..2..1....2..0..1..0..1....2..2..1..0..1
%e A206216 ..0..0..0..0..0....0..0..0..1..1....0..0..2..0..2....0..0..0..0..2
%e A206216 ..0..0..1..1..0....0..1..0..1..2....0..0..1..0..2....0..0..1..0..2
%K A206216 nonn,new
%O A206216 1,1
%A A206216 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A177505
%S A177505 0,1,2,3,304,305,306,307,288,289,290,291,272,273,274,275,256,257,258,
%T A177505 259,560,561,562,563,544,545,546,547,528,529,530,531,512,513,514,515,
%U A177505 816,817,818,819,800,801,802,803,784,785
%N A177505 Base 2i representation of n reinterpreted in base 4
%C A177505 The use of negabinary dispenses with the need for sign bits and for keeping track of signed and unsigned data types. Similarly, the use of base 2i, or quarter-imaginary, dispenses with the need to represent the real and imaginary parts of a complex number separately. (The term "quarter-imaginary" appears in Knuth's landmark book on computer programming).
%C A177505 Quarter-imaginary, based on the powers of 2i (twice the imaginary unit), uses the digits 0, 1, 2, 3. For purely real positive integers, the quarter-imaginary representation is the same as negaquartal (base -4) except that 0s are "riffled" in, corresponding to the odd-indexed powers of 2i which are purely imaginary numbers. Therefore, to obtain the base 2i representations of positive real numbers, the algorithm for base -4 representations can be employed with only a small adjustment.
%D A177505 Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (1981): 189
%H A177505 Vincenzo Librandi, <a href="/A177505/b177505.txt">Table of n, a(n) for n = 0..1000</a>
%e A177505 a(5) = 305 because 5 in base 2i is 10301 ( = (2i)^4 + 3 * (2i)^2 + (2i)^0), and (-4)^4 + 3 * (-4)^2 + (-4)^0 = 256 + 3 * 16 + 1 = 305.
%t A177505 (* First run the program from A039724 to define ToNegaBases *) Table[FromDigits[Riffle[IntegerDigits[ToNegaBases[n, 4]], 0], 4], {n, 0, 63}]
%Y A177505 Cf. A005351, base -2 representation of n reinterpreted as binary.
%K A177505 nonn,easy,base,new
%O A177505 0,3
%A A177505 Alonso del Arte (alonso.delarte(AT)gmail.com), Feb 03 2012

%I A206215
%S A206215 14,45,45,162,130,162,594,336,336,594,2268,992,760,992,2268,8802,2996,
%T A206215 2224,2224,2996,8802,34236,9072,5632,8136,5632,9072,34236,133974,
%U A206215 27656,14416,21984,21984,14416,27656,133974,525636,84152,40032,74080,86152
%N A206215 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206215 Table starts
%C A206215 .....14....45....162....594....2268.....8802.....34236.....133974......525636
%C A206215 .....45...130....336....992....2996.....9072.....27656......84152......256416
%C A206215 ....162...336....760...2224....5632....14416.....40032.....101376......259488
%C A206215 ....594...992...2224...8136...21984....74080....274320.....747456.....2518720
%C A206215 ...2268..2996...5632..21984...86152...359424...1435584....5673232....23721984
%C A206215 ...8802..9072..14416..74080..359424..1764000...9438912...46540800...228988224
%C A206215 ..34236.27656..40032.274320.1435584..9438912..68074272..366583680..2427719040
%C A206215 .133974.84152.101376.747456.5673232.46540800.366583680.2870491680.23806365696
%H A206215 R. H. Hardin, <a href="/A206215/b206215.txt">Table of n, a(n) for n = 1..840</a>
%F A206215 Empirical for column k:
%F A206215 k=1: a(n) = 4*a(n-1) -a(n-2) +12*a(n-3) -36*a(n-4) for n>5
%F A206215 k=2: a(n) = a(n-1) +6*a(n-2) +4*a(n-3) -10*a(n-4) for n>6
%F A206215 k=3: a(n) = 18*a(n-3) for n>6
%F A206215 k=4: a(n) = 34*a(n-3) for n>7
%F A206215 k=5: a(n) = 66*a(n-3) for n>8
%F A206215 k=6: a(n) = 130*a(n-3) for n>9
%F A206215 k=7: a(n) = 258*a(n-3) for n>10
%F A206215 apparently a(n) = (2^(k+1) +2)*a(n-3) for k>2 and n>k+3
%e A206215 Some solutions for n=4 k=3
%e A206215 ..0..1..0..2....0..0..1..2....0..0..1..2....0..0..1..0....0..0..1..1
%e A206215 ..1..0..0..2....2..0..0..1....0..1..2..2....0..1..0..0....2..2..0..1
%e A206215 ..0..0..1..0....0..2..0..0....1..2..2..0....1..0..0..2....0..2..2..0
%e A206215 ..0..2..0..0....0..0..1..0....2..2..0..1....0..0..1..0....1..0..2..2
%e A206215 ..2..0..0..1....1..0..0..1....2..0..1..1....0..1..0..0....1..1..0..0
%K A206215 nonn,tabl,new
%O A206215 1,1
%A A206215 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206214
%S A206214 34236,27656,40032,274320,1435584,9438912,68074272,366583680,
%T A206214 2427719040,17541854784,94578589440,626351512320,4525798534272,
%U A206214 24401276075520,161598690178560,1167656021842176,6295529227484160,41692462066068480
%N A206214 Number of (n+1)X8 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206214 Column 7 of A206215
%H A206214 R. H. Hardin, <a href="/A206214/b206214.txt">Table of n, a(n) for n = 1..210</a>
%F A206214 Empirical: a(n) = 258*a(n-3) for n>10
%e A206214 Some solutions for n=4
%e A206214 ..0..1..0..2..1..1..2..1....0..1..0..0..1..2..1..2....0..1..1..2..0..0..1..0
%e A206214 ..1..1..0..0..2..1..1..2....1..0..0..1..2..2..1..1....1..1..2..0..0..1..0..0
%e A206214 ..2..2..1..0..0..2..1..1....0..0..1..2..2..0..2..2....1..2..0..0..1..0..0..1
%e A206214 ..0..2..2..1..0..0..2..1....0..1..2..2..0..2..2..0....2..0..0..1..0..0..1..0
%e A206214 ..1..0..2..2..1..0..0..2....1..2..2..1..2..2..0..2....0..0..2..0..0..1..0..0
%K A206214 nonn,new
%O A206214 1,1
%A A206214 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206213
%S A206213 8802,9072,14416,74080,359424,1764000,9438912,46540800,228988224,
%T A206213 1227058560,6050304000,29768469120,159517612800,786539520000,
%U A206213 3869900985600,20737289664000,102250137600000,503087128128000
%N A206213 Number of (n+1)X7 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206213 Column 6 of A206215
%H A206213 R. H. Hardin, <a href="/A206213/b206213.txt">Table of n, a(n) for n = 1..210</a>
%F A206213 Empirical: a(n) = 130*a(n-3) for n>9
%e A206213 Some solutions for n=4
%e A206213 ..0..1..0..2..1..1..0....0..0..1..2..2..1..1....0..1..1..2..1..1..2
%e A206213 ..1..1..0..0..2..1..1....0..1..2..2..0..2..2....2..0..1..1..0..1..1
%e A206213 ..0..0..2..0..0..2..1....1..2..2..0..2..2..0....2..2..0..1..1..2..1
%e A206213 ..1..0..0..2..0..0..2....2..2..0..2..2..1..2....0..2..2..0..1..1..0
%e A206213 ..2..1..0..0..2..0..2....0..0..2..2..1..2..2....1..0..2..2..0..1..0
%K A206213 nonn,new
%O A206213 1,1
%A A206213 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206212
%S A206212 2268,2996,5632,21984,86152,359424,1435584,5673232,23721984,94748544,
%T A206212 374433312,1565650944,6253403904,24712598592,103332962304,
%U A206212 412724657664,1631031507072,6819975512064,27239827405824,107648079466752
%N A206212 Number of (n+1)X6 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206212 Column 5 of A206215
%H A206212 R. H. Hardin, <a href="/A206212/b206212.txt">Table of n, a(n) for n = 1..210</a>
%F A206212 Empirical: a(n) = 66*a(n-3) for n>810</a>
%e A206212 Some solutions for n=4
%e A206212 ..0..0..1..1..0..2....0..1..2..0..1..1....0..0..1..0..2..1....0..0..1..0..2..0
%e A206212 ..1..1..0..1..1..0....1..1..2..2..0..1....0..1..0..0..2..2....0..2..0..0..2..2
%e A206212 ..2..1..1..2..1..1....0..0..1..2..2..0....1..0..0..2..1..1....1..0..0..1..0..0
%e A206212 ..1..0..1..1..2..2....1..0..0..1..2..2....0..0..2..1..1..0....0..0..1..0..0..1
%e A206212 ..1..1..2..1..2..0....2..1..0..0..1..1....2..2..1..1..0..1....2..2..0..0..1..0
%K A206212 nonn,new
%O A206212 1,1
%A A206212 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206211
%S A206211 594,992,2224,8136,21984,74080,274320,747456,2518720,9326880,25413504,
%T A206211 85636480,317113920,864059136,2911640320,10781873280,29378010624,
%U A206211 98995770880,366583691520,998852361216,3365856209920,12463845511680
%N A206211 Number of (n+1)X5 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206211 Column 4 of A206215
%H A206211 R. H. Hardin, <a href="/A206211/b206211.txt">Table of n, a(n) for n = 1..210</a>
%F A206211 Empirical: a(n) = 34*a(n-3) for n>7
%e A206211 Some solutions for n=4
%e A206211 ..0..0..1..0..1....0..0..1..2..2....0..0..1..0..1....0..0..1..0..2
%e A206211 ..0..1..0..0..1....0..1..2..2..0....0..2..0..0..1....0..1..0..0..2
%e A206211 ..2..0..0..1..2....1..2..2..0..2....1..0..0..1..0....2..0..0..1..0
%e A206211 ..0..0..1..2..2....2..2..0..2..2....0..0..1..0..0....0..0..1..0..0
%e A206211 ..0..1..2..2..0....2..1..2..2..0....2..2..0..0..1....1..1..0..0..1
%K A206211 nonn,new
%O A206211 1,1
%A A206211 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206210
%S A206210 162,336,760,2224,5632,14416,40032,101376,259488,720576,1824768,
%T A206210 4670784,12970368,32845824,84074112,233466624,591224832,1513334016,
%U A206210 4202399232,10642046976,27240012288,75643186176,191556845568,490320221184
%N A206210 Number of (n+1)X4 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206210 Column 3 of A206215
%H A206210 R. H. Hardin, <a href="/A206210/b206210.txt">Table of n, a(n) for n = 1..210</a>
%F A206210 Empirical: a(n) = 18*a(n-3) for n>6
%e A206210 Some solutions for n=4
%e A206210 ..0..1..2..2....0..1..1..0....0..0..1..2....0..0..1..1....0..1..0..2
%e A206210 ..1..2..2..0....2..0..1..1....2..0..0..1....2..2..0..1....1..0..0..2
%e A206210 ..2..2..0..1....2..2..0..1....1..2..0..0....0..2..2..0....0..0..1..0
%e A206210 ..2..0..1..1....0..2..2..0....1..1..2..2....1..0..2..2....0..2..0..0
%e A206210 ..0..1..1..2....2..1..2..2....2..1..2..0....1..1..0..0....2..0..0..1
%K A206210 nonn,new
%O A206210 1,1
%A A206210 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206209
%S A206209 45,130,336,992,2996,9072,27656,84152,256416,781232,2379776,7251312,
%T A206209 22090736,67305392,205057296,624739472,1903397456,5799009552,
%U A206209 17667779216,53828031632,163996770576,499645981712,1522260939536,4637843595792
%N A206209 Number of (n+1)X3 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206209 Column 2 of A206215
%H A206209 R. H. Hardin, <a href="/A206209/b206209.txt">Table of n, a(n) for n = 1..210</a>
%F A206209 Empirical: a(n) = a(n-1) +6*a(n-2) +4*a(n-3) -10*a(n-4) for n>6
%e A206209 Some solutions for n=4
%e A206209 ..0..0..0....0..1..2....0..1..1....0..1..0....0..0..1....0..1..1....0..1..2
%e A206209 ..1..2..1....0..0..1....0..2..0....2..0..0....0..2..0....1..2..1....0..0..1
%e A206209 ..1..0..1....1..0..0....0..1..0....0..0..1....2..0..0....1..1..2....1..0..0
%e A206209 ..1..2..1....0..2..0....0..2..0....0..1..2....0..0..2....2..1..1....2..1..1
%e A206209 ..0..0..0....0..0..1....0..1..0....1..2..2....0..2..1....1..2..2....1..1..2
%K A206209 nonn,new
%O A206209 1,1
%A A206209 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206208
%S A206208 14,45,162,594,2268,8802,34236,133974,525636,2062530,8099676,31820742,
%T A206208 125010756,491167314,1929919068,7583091318,29796066756,117078181218,
%U A206208 460036667484,1807630002342,7102763113284,27909047936754,109663668632412
%N A206208 Number of (n+1)X2 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206208 Column 1 of A206215
%H A206208 R. H. Hardin, <a href="/A206208/b206208.txt">Table of n, a(n) for n = 1..210</a>
%F A206208 Empirical: a(n) = 4*a(n-1) -a(n-2) +12*a(n-3) -36*a(n-4) for n>5
%e A206208 Some solutions for n=4
%e A206208 ..0..0....0..0....0..1....0..0....0..0....0..1....0..1....0..0....0..1....0..1
%e A206208 ..0..1....0..1....1..0....0..1....0..1....0..2....0..2....0..1....2..0....1..2
%e A206208 ..1..2....1..0....1..0....1..2....2..0....0..1....1..2....1..2....0..0....1..1
%e A206208 ..1..2....1..0....2..1....1..2....2..2....2..1....0..2....1..1....0..1....2..1
%e A206208 ..2..0....0..1....1..1....0..1....1..1....0..1....0..1....0..0....2..0....1..2
%K A206208 nonn,new
%O A206208 1,1
%A A206208 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206207
%S A206207 14,130,760,8136,86152,1764000,68074272,2870491680,232945511040,
%T A206207 35774121536640,6032245277059200,1955894774772902400,
%U A206207 1200725012024612467200,809841945662029576512000
%N A206207 Number of (n+1)X(n+1) 0..2 arrays with every 2X3 or 3X2 subblock having exactly two equal edges, and new values 0..2 introduced in row major order
%C A206207 Diagonal of A206215
%H A206207 R. H. Hardin, <a href="/A206207/b206207.txt">Table of n, a(n) for n = 1..20</a>
%e A206207 Some solutions for n=4
%e A206207 ..0..0..1..0..1....0..0..1..0..1....0..1..0..0..2....0..0..1..0..1
%e A206207 ..0..1..0..0..1....0..2..0..0..1....0..0..2..0..0....0..2..0..0..1
%e A206207 ..1..0..0..1..2....2..0..0..2..0....1..0..0..2..0....1..0..0..2..0
%e A206207 ..0..0..1..2..2....0..0..1..0..0....2..1..0..0..1....0..0..1..0..0
%e A206207 ..0..1..2..2..0....2..2..0..0..2....2..2..1..0..0....2..2..0..0..1
%K A206207 nonn,new
%O A206207 1,1
%A A206207 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206206
%S A206206 256,480,480,1768,1060,1768,7388,3008,3008,7388,29724,9464,5136,9464,
%T A206206 29724,117084,22576,11832,11832,22576,117084,468968,67968,27400,32920,
%U A206206 27400,67968,468968,1878808,192808,60800,70168,70168,60800,192808
%N A206206 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206206 Table starts
%C A206206 .....256....480...1768...7388...29724..117084..468968..1878808..7493008
%C A206206 .....480...1060...3008...9464...22576...67968..192808...496088..1475768
%C A206206 ....1768...3008...5136..11832...27400...60800..145208...346080...775944
%C A206206 ....7388...9464..11832..32920...70168..109720..337896...728704..1137208
%C A206206 ...29724..22576..27400..70168..118936..236488..713408..1226384..2475328
%C A206206 ..117084..67968..60800.109720..236488..484704.1080648..2383520..5023520
%C A206206 ..468968.192808.145208.337896..713408.1080648.3270240..7008160.10936160
%C A206206 .1878808.496088.346080.728704.1226384.2383520.7008160.11893600.23648800
%H A206206 R. H. Hardin, <a href="/A206206/b206206.txt">Table of n, a(n) for n = 1..759</a>
%F A206206 Empirical for column k:
%F A206206 k=1: a(n) = 3*a(n-1) +17*a(n-3) -5*a(n-4) +2*a(n-5) for n>6
%F A206206 k=2: a(n) = 3*a(n-2) +14*a(n-3) -3*a(n-4) +a(n-5) +a(n-6) +a(n-7) for n>9
%F A206206 k=3: a(n) = 10*a(n-3) +5*a(n-4) +a(n-5) +2*a(n-8) +2*a(n-10) for n>18
%F A206206 k=4: a(n) = 10*a(n-3) +a(n-9) for n>18
%F A206206 k=5: a(n) = 10*a(n-3) for n>15
%F A206206 k=6: a(n) = 10*a(n-3) for n>16
%F A206206 k=7: a(n) = 10*a(n-3) for n>17
%F A206206 apparently a(n) = 10*a(n-3) for k>4 and n>k+5
%e A206206 Some solutions for n=4 k=3
%e A206206 ..3..3..3..2....1..3..2..2....1..0..2..2....3..2..2..0....3..0..0..0
%e A206206 ..3..0..0..2....1..1..2..2....1..1..2..2....0..2..2..2....3..3..0..0
%e A206206 ..0..0..0..1....1..1..1..2....1..1..1..2....0..0..2..2....3..3..3..0
%e A206206 ..0..0..1..1....0..1..1..1....3..1..1..1....0..0..0..2....0..3..3..0
%e A206206 ..1..1..1..1....0..0..1..1....3..3..1..1....2..0..0..1....0..0..0..0
%K A206206 nonn,tabl,new
%O A206206 1,1
%A A206206 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206205
%S A206205 468968,192808,145208,337896,713408,1080648,3270240,7008160,10936160,
%T A206205 33660800,72736960,113216800,339966080,729156480,1132733728,
%U A206205 3399739520,7291568000,11327337280,33997395200,72915680000,113273372800
%N A206205 Number of (n+1)X8 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206205 Column 7 of A206206
%H A206205 R. H. Hardin, <a href="/A206205/b206205.txt">Table of n, a(n) for n = 1..210</a>
%F A206205 Empirical: a(n) = 10*a(n-3) for n>17
%e A206205 Some solutions for n=4
%e A206205 ..1..1..2..2..0..3..3..1....2..0..3..3..0..0..0..1....0..1..1..3..3..3..2..0
%e A206205 ..1..2..2..2..3..3..3..1....2..2..3..3..3..0..0..0....0..1..1..1..3..3..3..0
%e A206205 ..2..2..2..3..3..3..1..1....2..2..2..3..3..3..0..0....0..0..1..1..1..3..3..3
%e A206205 ..2..2..3..3..3..1..1..1....3..2..2..2..3..3..3..0....0..0..0..1..1..1..3..3
%e A206205 ..2..3..3..3..0..1..1..3....0..0..2..2..2..3..3..0....1..0..0..0..1..1..1..1
%K A206205 nonn,new
%O A206205 1,1
%A A206205 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206204
%S A206204 117084,67968,60800,109720,236488,484704,1080648,2383520,5023520,
%T A206204 11272288,24906080,51970688,113273344,249152672,519714752,1132733728,
%U A206204 2491526720,5197147520,11327337280,24915267200,51971475200,113273372800
%N A206204 Number of (n+1)X7 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206204 Column 6 of A206206
%H A206204 R. H. Hardin, <a href="/A206204/b206204.txt">Table of n, a(n) for n = 1..210</a>
%F A206204 Empirical: a(n) = 10*a(n-3) for n>16
%e A206204 Some solutions for n=4
%e A206204 ..0..3..1..1..2..2..2....1..0..0..0..0..3..1....3..0..0..3..1..1..1
%e A206204 ..0..0..1..1..1..2..2....2..3..3..0..0..0..1....0..0..0..1..1..1..3
%e A206204 ..0..0..0..1..1..1..2....2..3..3..3..0..0..0....0..0..1..1..1..2..2
%e A206204 ..1..0..0..0..1..1..1....2..2..3..3..3..0..0....0..1..1..1..2..2..2
%e A206204 ..2..2..0..0..0..1..1....2..2..2..3..3..3..0....0..1..1..2..2..2..1
%K A206204 nonn,new
%O A206204 1,1
%A A206204 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206203
%S A206203 29724,22576,27400,70168,118936,236488,713408,1226384,2475328,7286528,
%T A206203 12463904,24915264,72915680,124641664,249152672,729156800,1246416640,
%U A206203 2491526720,7291568000,12464166400,24915267200,72915680000,124641664000
%N A206203 Number of (n+1)X6 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206203 Column 5 of A206206
%H A206203 R. H. Hardin, <a href="/A206203/b206203.txt">Table of n, a(n) for n = 1..210</a>
%F A206203 Empirical: a(n) = 10*a(n-3) for n>15
%e A206203 Some solutions for n=4
%e A206203 ..1..1..2..2..1..0....3..0..0..1..1..1....0..0..2..1..1..0....3..0..0..1..1..1
%e A206203 ..1..2..2..2..3..3....3..0..0..0..1..1....0..0..1..1..1..0....3..0..0..0..1..1
%e A206203 ..2..2..2..3..3..3....3..3..0..0..0..1....0..1..1..1..3..3....3..3..0..0..0..1
%e A206203 ..2..2..3..3..3..0....3..3..3..0..0..0....1..1..1..3..3..3....3..3..3..0..0..0
%e A206203 ..2..3..3..3..0..0....2..3..3..3..0..0....1..1..3..3..3..1....0..3..3..3..0..0
%K A206203 nonn,new
%O A206203 1,1
%A A206203 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206202
%S A206202 7388,9464,11832,32920,70168,109720,337896,728704,1137208,3418192,
%T A206202 7313872,11389472,34215016,73207664,114003240,342487240,732805056,
%U A206202 1141169536,3428290592,7335364432,11423084832,34317120936,73426851984
%N A206202 Number of (n+1)X5 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206202 Column 4 of A206206
%H A206202 R. H. Hardin, <a href="/A206202/b206202.txt">Table of n, a(n) for n = 1..210</a>
%F A206202 Empirical: a(n) = 10*a(n-3) +a(n-9) for n>18
%e A206202 Some solutions for n=4
%e A206202 ..0..1..1..2..2....1..0..0..0..0....3..0..0..3..2....2..3..3..2..2
%e A206202 ..0..1..1..1..2....0..0..0..2..2....3..0..0..0..1....2..3..3..3..1
%e A206202 ..0..0..1..1..1....0..0..2..2..2....3..3..0..0..0....2..2..3..3..3
%e A206202 ..0..0..0..1..1....0..2..2..2..1....3..3..3..0..0....2..2..2..3..3
%e A206202 ..1..0..0..0..0....1..2..2..0..1....1..3..3..3..3....1..2..2..2..2
%K A206202 nonn,new
%O A206202 1,1
%A A206202 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206201
%S A206201 1768,3008,5136,11832,27400,60800,145208,346080,775944,1799656,
%T A206201 4265112,9666256,22287416,52569856,120132224,276283624,648644008,
%U A206201 1490750392,3426149328,8010922248,18480112640,42488361472,99015558792,228939593928
%N A206201 Number of (n+1)X4 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206201 Column 3 of A206206
%H A206201 R. H. Hardin, <a href="/A206201/b206201.txt">Table of n, a(n) for n = 1..210</a>
%F A206201 Empirical: a(n) = 10*a(n-3) +5*a(n-4) +a(n-5) +2*a(n-8) +2*a(n-10) for n>18
%e A206201 Some solutions for n=4
%e A206201 ..0..3..1..1....0..2..1..1....3..3..1..0....0..0..2..2....1..0..0..0
%e A206201 ..0..1..1..1....0..1..1..1....3..3..3..0....0..0..1..1....0..0..0..1
%e A206201 ..1..1..1..3....1..1..1..0....1..3..3..3....0..1..1..1....0..0..1..1
%e A206201 ..1..1..2..2....1..1..2..2....2..2..3..3....0..1..1..0....0..1..1..1
%e A206201 ..1..2..2..2....1..2..2..2....2..2..2..2....0..0..0..0....0..1..1..0
%K A206201 nonn,new
%O A206201 1,1
%A A206201 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206200
%S A206200 480,1060,3008,9464,22576,67968,192808,496088,1475768,4018720,
%T A206200 10894120,31512000,85274000,237162184,670298216,1827175160,5121768064,
%U A206200 14281894072,39288809968,110061644032,305182278280,845001830872,2359774258680
%N A206200 Number of (n+1)X3 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206200 Column 2 of A206206
%H A206200 R. H. Hardin, <a href="/A206200/b206200.txt">Table of n, a(n) for n = 1..210</a>
%F A206200 Empirical: a(n) = 3*a(n-2) +14*a(n-3) -3*a(n-4) +a(n-5) +a(n-6) +a(n-7) for n>9
%e A206200 Some solutions for n=4
%e A206200 ..0..0..1....1..2..2....3..0..0....1..0..2....3..0..0....0..3..3....0..1..1
%e A206200 ..0..0..1....2..2..2....3..3..0....1..2..2....3..3..0....3..3..3....1..1..1
%e A206200 ..0..1..1....2..2..3....3..3..3....1..2..2....3..3..3....3..3..0....1..1..2
%e A206200 ..0..1..1....2..3..3....1..3..3....1..1..2....2..3..3....3..0..0....1..2..2
%e A206200 ..0..0..1....2..3..3....1..1..3....1..1..2....2..2..2....3..0..0....1..2..2
%K A206200 nonn,new
%O A206200 1,1
%A A206200 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206199
%S A206199 256,480,1768,7388,29724,117084,468968,1878808,7493008,29925508,
%T A206199 119605588,477741796,1908251600,7623408272,30453658424,121651754668,
%U A206199 485967430220,1941313945708,7755000190904,30979095429960,123753089725152
%N A206199 Number of (n+1)X2 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206199 Column 1 of A206206
%H A206199 R. H. Hardin, <a href="/A206199/b206199.txt">Table of n, a(n) for n = 1..210</a>
%F A206199 Empirical: a(n) = 3*a(n-1) +17*a(n-3) -5*a(n-4) +2*a(n-5) for n>6
%e A206199 Some solutions for n=4
%e A206199 ..1..3....1..2....2..0....1..1....0..2....1..3....1..0....3..3....1..0....2..2
%e A206199 ..1..1....1..2....2..2....1..3....0..0....2..2....0..0....1..1....0..0....1..1
%e A206199 ..1..1....1..2....2..2....3..3....0..0....2..2....0..0....1..1....0..0....1..1
%e A206199 ..3..0....1..1....0..0....3..3....1..0....2..0....0..2....2..3....1..2....2..0
%e A206199 ..3..3....1..1....0..0....2..2....1..0....2..3....1..1....2..2....1..1....3..3
%K A206199 nonn,new
%O A206199 1,1
%A A206199 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206198
%S A206198 256,1060,5136,32920,118936,484704,3270240,11893600,48470400,
%T A206198 327024000,1189360000,4847040000,32702400000,118936000000,
%U A206198 484704000000,3270240000000,11893600000000,48470400000000,327024000000000
%N A206198 Number of (n+1)X(n+1) 0..3 arrays with every 2X3 or 3X2 subblock having exactly one clockwise edge increases
%C A206198 Diagonal of A206206
%e A206198 Some solutions for n=4
%e A206198 ..3..3..2..0..0....2..3..3..2..2....2..2..2..1..0....0..1..1..2..2
%e A206198 ..3..3..3..0..0....2..3..3..3..1....3..2..2..2..0....0..1..1..1..2
%e A206198 ..0..3..3..3..0....2..2..3..3..3....1..1..2..2..2....0..0..1..1..1
%e A206198 ..0..0..3..3..3....2..2..2..3..3....1..1..1..2..2....0..0..0..1..1
%e A206198 ..0..0..0..3..3....1..2..2..2..2....2..1..1..1..1....1..0..0..0..0
%K A206198 nonn,new
%O A206198 1,1
%A A206198 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206197
%S A206197 124,1012,1012,8304,18172,8304,68180,329216,329216,68180,559788,
%T A206197 5972392,13173864,5972392,559788,4596072,108363344,527945420,
%U A206197 527945420,108363344,4596072,37735460,1966171084,21162670972,46704516192,21162670972
%N A206197 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206197 Table starts
%C A206197 .......124.........1012.............8304...............68180
%C A206197 ......1012........18172...........329216.............5972392
%C A206197 ......8304.......329216.........13173864...........527945420
%C A206197 .....68180......5972392........527945420.........46704516192
%C A206197 ....559788....108363344......21162670972.......4131608458140
%C A206197 ...4596072...1966171084.....848327900384.....365468694266436
%C A206197 ..37735460..35674679732...34006136221640...32327015669963752
%C A206197 .309822148.647289744976.1363171550451384.2859398851677838652
%H A206197 R. H. Hardin, <a href="/A206197/b206197.txt">Table of n, a(n) for n = 1..144</a>
%e A206197 Some solutions for n=4 k=3
%e A206197 ..2..2..1..2....1..0..0..0....1..1..2..0....3..0..1..0....2..3..1..1
%e A206197 ..0..3..2..0....0..3..2..1....3..2..0..3....1..2..2..1....2..1..3..2
%e A206197 ..2..0..3..1....1..0..3..2....0..3..1..0....0..1..0..3....3..2..0..3
%e A206197 ..3..2..0..2....3..1..0..3....1..0..2..1....2..2..1..0....0..3..1..0
%e A206197 ..0..3..1..0....0..2..1..0....3..2..3..2....0..3..2..1....1..0..2..1
%K A206197 nonn,tabl,new
%O A206197 1,1
%A A206197 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206196
%S A206196 37735460,35674679732,34006136221640,32327015669963752,
%T A206196 30663225851582320716,29048689240586916141284,
%U A206196 27501116060342502948845780,26027294273532996135747769404,24628320993481612254311730982740
%N A206196 Number of (n+1)X8 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206196 Column 7 of A206197
%H A206196 R. H. Hardin, <a href="/A206196/b206196.txt">Table of n, a(n) for n = 1..167</a>
%e A206196 Some solutions for n=4
%e A206196 ..3..2..3..1..2..2..2..1....2..2..2..1..2..0..1..0....3..2..0..2..2..3..0..0
%e A206196 ..2..3..0..3..1..0..3..2....1..0..3..2..3..2..2..1....0..3..1..0..3..2..2..1
%e A206196 ..3..0..2..0..2..1..0..3....3..2..0..3..1..0..3..2....1..0..3..2..1..0..3..2
%e A206196 ..2..2..3..1..3..2..1..0....0..3..1..0..2..1..0..3....0..1..0..3..2..1..0..3
%e A206196 ..1..0..2..3..2..1..0..3....1..0..2..1..3..2..1..0....1..2..1..0..3..2..1..0
%K A206196 nonn,new
%O A206196 1,1
%A A206196 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206195
%S A206195 4596072,1966171084,848327900384,365468694266436,157233431036489148,
%T A206195 67595116900832029672,29048689240586916141284,
%U A206195 12481384538302527859923440,5362463708438175275453119020,2303828616153300698234374000660
%N A206195 Number of (n+1)X7 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206195 Column 6 of A206197
%H A206195 R. H. Hardin, <a href="/A206195/b206195.txt">Table of n, a(n) for n = 1..210</a>
%e A206195 Some solutions for n=4
%e A206195 ..2..0..0..1..0..3..2....3..1..3..2..0..2..2....2..2..0..1..2..0..3
%e A206195 ..3..2..1..2..1..0..3....3..2..0..3..1..0..3....0..3..1..0..2..1..0
%e A206195 ..0..3..2..0..2..1..0....0..3..2..1..2..1..0....1..0..2..1..3..2..1
%e A206195 ..1..0..3..1..3..2..1....3..0..3..2..3..2..1....3..1..3..2..0..3..2
%e A206195 ..3..1..0..2..1..3..2....3..1..0..3..1..0..2....2..1..0..3..1..0..3
%K A206195 nonn,new
%O A206195 1,1
%A A206195 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206194
%S A206194 559788,108363344,21162670972,4131608458140,806129409968360,
%T A206194 157233431036489148,30663225851582320716,5979459481371854518796,
%U A206194 1165987978329531107899148,227363834100630151941061428
%N A206194 Number of (n+1)X6 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206194 Column 5 of A206197
%H A206194 R. H. Hardin, <a href="/A206194/b206194.txt">Table of n, a(n) for n = 1..210</a>
%e A206194 Some solutions for n=4
%e A206194 ..0..2..2..2..3..1....1..0..2..0..3..3....3..2..0..0..3..3....1..0..0..3..2..3
%e A206194 ..2..1..0..3..1..0....0..3..3..2..1..0....1..0..3..2..1..0....3..2..1..0..3..0
%e A206194 ..0..2..1..0..2..1....2..1..0..3..2..1....2..1..0..3..2..1....0..3..2..1..0..2
%e A206194 ..1..3..2..1..3..2....0..2..1..0..3..2....1..3..2..0..3..2....2..1..0..3..2..3
%e A206194 ..2..0..3..2..0..3....1..0..2..1..0..3....0..0..3..2..0..3....0..2..1..0..3..1
%K A206194 nonn,new
%O A206194 1,1
%A A206194 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206193
%S A206193 68180,5972392,527945420,46704516192,4131608458140,365468694266436,
%T A206193 32327015669963752,2859398851677838652,252919077269469177332,
%U A206193 22371116614243605962696,1978761607489986202200072
%N A206193 Number of (n+1)X5 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206193 Column 4 of A206197
%H A206193 R. H. Hardin, <a href="/A206193/b206193.txt">Table of n, a(n) for n = 1..210</a>
%F A206193 Empirical: a(n) = 182*a(n-1) -12004*a(n-2) +415480*a(n-3) -8853859*a(n-4) +126860780*a(n-5) -1289524547*a(n-6) +9523743134*a(n-7) -51068777938*a(n-8) +192111360492*a(n-9) -439536805765*a(n-10) +123375390008*a(n-11) +3350213610392*a(n-12) -13024192131153*a(n-13) +15930041442264*a(n-14) +62368141603379*a(n-15) -366404617203830*a(n-16) +752070396824663*a(n-17) +183801635239481*a(n-18) -5068022786445445*a(n-19) +14232658005953631*a(n-20) -27239219734522543*a(n-21) +67543448390055318*a(n-22) -189380456822496217*a(n-23) +359698186059711841*a(n-24) -352001537255358915*a(n-25) +149211548566299027*a(n-26) -482172263712426285*a(n-27) +1837850024880558262*a(n-28) -2536427093507181031*a(n-29) -76797065651444300*a(n-30) +3296000796319362439*a(n-31) +619616134205467421*a(n-32) -12492584079381734747*a(n-33) +18456044380698042369*a(n-34) -6277092897298636065*a(n-35) -14904702224653191032*a(n-36) +22060151606971342463*a(n-37) -7648135521787651695*a(n-38) -8849848562303425202*a(n-39) +10019643948658059552*a(n-40) +205970292858437768*a(n-41) -7836482072410929685*a(n-42) +6998666575490914763*a(n-43) -2332335436415594915*a(n-44) -767081781945942946*a(n-45) +1695746347742151173*a(n-46) -860795289270765768*a(n-47) -656278600351921584*a(n-48) +491914658871178517*a(n-49) +157475347940078023*a(n-50) -195175378597420476*a(n-51) -3952840614780389*a(n-52) +17786104824460301*a(n-53) +24965620043420133*a(n-54) +801930069620387*a(n-55) -11862045800221349*a(n-56) +37878482250943*a(n-57) +701917001215691*a(n-58) +970284308515894*a(n-59) +65259701649948*a(n-60) -96105288061917*a(n-61) -51798674181470*a(n-62) -15738036525022*a(n-63) +5269455880264*a(n-64) +1290286596342*a(n-65) +768172777163*a(n-66) -51366586551*a(n-67) -11887411470*a(n-68) -4642143720*a(n-69) -6591574780*a(n-70) +415584680*a(n-71) +59348212*a(n-72) +6709172*a(n-73) +11723720*a(n-74) -243696*a(n-75) -112288*a(n-76) -4992*a(n-77) -1024*a(n-78)
%e A206193 Some solutions for n=4
%e A206193 ..1..2..1..1..0....3..2..1..0..2....3..2..3..1..1....2..2..1..0..0
%e A206193 ..0..1..0..3..2....2..3..2..1..3....2..0..1..0..2....1..0..3..2..1
%e A206193 ..1..2..1..0..3....0..0..3..2..1....0..1..3..1..3....2..1..0..3..2
%e A206193 ..1..3..2..1..0....3..1..0..3..2....1..2..0..2..0....3..2..1..0..3
%e A206193 ..1..0..3..2..1....0..2..1..0..3....1..0..3..2..1....0..3..2..1..0
%K A206193 nonn,new
%O A206193 1,1
%A A206193 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206192
%S A206192 8304,329216,13173864,527945420,21162670972,848327900384,
%T A206192 34006136221640,1363171550451384,54644142459882172,
%U A206192 2190466837748368020,87807121912310588320,3519839010271451725584,141096375518760876476256
%N A206192 Number of (n+1)X4 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206192 Column 3 of A206197
%H A206192 R. H. Hardin, <a href="/A206192/b206192.txt">Table of n, a(n) for n = 1..210</a>
%F A206192 Empirical: a(n) = 69*a(n-1) -1503*a(n-2) +16061*a(n-3) -100719*a(n-4) +413071*a(n-5) -1200419*a(n-6) +2350991*a(n-7) -2650016*a(n-8) +246039*a(n-9) +5103223*a(n-10) -8853227*a(n-11) +5015318*a(n-12) +4249674*a(n-13) -9106668*a(n-14) +7509128*a(n-15) -3026335*a(n-16) +952523*a(n-17) -1880226*a(n-18) -37992*a(n-19) -166501*a(n-20) -450312*a(n-21) -419109*a(n-22) +222765*a(n-23) +217432*a(n-24) -15270*a(n-25) +13404*a(n-26) +5852*a(n-27) +348*a(n-28) for n>29
%e A206192 Some solutions for n=4
%e A206192 ..2..3..3..0....1..1..2..0....1..0..2..1....2..1..1..2....1..3..3..3
%e A206192 ..1..2..0..2....3..2..0..3....0..1..0..2....0..3..2..0....3..2..1..0
%e A206192 ..3..1..1..3....0..3..1..0....2..2..1..3....3..0..3..2....0..3..2..1
%e A206192 ..0..3..2..0....1..0..2..1....0..3..2..0....1..3..2..3....2..0..3..2
%e A206192 ..3..0..3..1....3..2..3..2....1..0..3..1....2..0..3..2....2..1..0..3
%K A206192 nonn,new
%O A206192 1,1
%A A206192 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206191
%S A206191 1012,18172,329216,5972392,108363344,1966171084,35674679732,
%T A206191 647289744976,11744575881312,213096317372352,3866469156691484,
%U A206191 70154115855170420,1272892598372080320,23095659424144383792,419053017449035125152
%N A206191 Number of (n+1)X3 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206191 Column 2 of A206197
%H A206191 R. H. Hardin, <a href="/A206191/b206191.txt">Table of n, a(n) for n = 1..210</a>
%F A206191 Empirical: a(n) = 23*a(n-1) -97*a(n-2) +169*a(n-3) -151*a(n-4) +250*a(n-5) -178*a(n-6) +66*a(n-7) -64*a(n-8) +8*a(n-9) +4*a(n-10)
%e A206191 Some solutions for n=4
%e A206191 ..1..0..0....1..3..0....2..1..1....0..1..0....3..2..1....2..3..1....0..2..3
%e A206191 ..0..2..1....1..0..1....1..0..3....2..0..1....1..0..2....1..1..3....2..1..1
%e A206191 ..2..1..0....2..1..2....2..1..0....2..1..3....3..1..3....3..2..1....0..3..2
%e A206191 ..1..3..1....1..3..3....0..2..1....0..2..1....0..2..0....0..3..2....1..0..3
%e A206191 ..3..0..2....0..1..0....1..3..2....3..3..2....1..3..2....3..1..3....3..2..0
%K A206191 nonn,new
%O A206191 1,1
%A A206191 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206190
%S A206190 124,1012,8304,68180,559788,4596072,37735460,309822148,2543754976,
%T A206190 20885173732,171475038268,1407873792488,11559167069396,94905057577684,
%U A206190 779205793959872,6397569158457620,52526420432849004,431261433077432552
%N A206190 Number of (n+1)X2 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206190 Column 1 of A206197
%H A206190 R. H. Hardin, <a href="/A206190/b206190.txt">Table of n, a(n) for n = 1..210</a>
%F A206190 Empirical: a(n) = 9*a(n-1) -7*a(n-2) +4*a(n-3) +2*a(n-4)
%e A206190 Some solutions for n=4
%e A206190 ..2..0....1..1....3..3....2..3....0..1....3..0....2..3....3..3....2..0....0..1
%e A206190 ..1..0....3..2....1..0....3..0....0..2....1..0....0..1....1..0....2..1....0..2
%e A206190 ..0..1....0..3....2..1....1..2....1..3....0..3....2..0....2..1....1..2....3..2
%e A206190 ..0..3....1..0....1..3....3..2....3..0....2..0....0..1....0..2....0..1....1..3
%e A206190 ..1..0....2..1....1..0....0..3....0..3....1..1....3..0....0..3....1..2....0..3
%K A206190 nonn,new
%O A206190 1,1
%A A206190 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206189
%S A206189 124,18172,13173864,46704516192,806129409968360,67595116900832029672,
%T A206189 27501116060342502948845780,54245028887229452020237690008284
%N A206189 Number of (n+1)X(n+1) 0..3 arrays with the number of clockwise edge increases in every 2X2 subblock equal to two
%C A206189 Diagonal of A206197
%e A206189 Some solutions for n=4
%e A206189 ..1..3..3..2..0....3..0..1..1..1....0..1..0..3..2....3..2..3..1..1
%e A206189 ..2..1..0..3..1....1..2..0..3..2....2..2..1..0..3....2..0..1..0..2
%e A206189 ..3..2..1..0..2....1..3..1..0..3....0..3..2..1..0....0..1..3..1..3
%e A206189 ..0..3..2..1..3....3..0..2..1..0....2..1..3..2..1....1..2..0..2..0
%e A206189 ..3..0..3..2..0....1..1..3..2..1....1..0..2..0..3....1..0..3..2..1
%K A206189 nonn,new
%O A206189 1,1
%A A206189 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206188
%S A206188 256,1976,1976,15616,19688,15616,124048,205804,205804,124048,986388,
%T A206188 2410188,3802676,2410188,986388,7854572,30417316,95187756,95187756,
%U A206188 30417316,7854572,62623348,406202048,2674444796,5674969344,2674444796
%N A206188 T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with every 2X2 subblock having the same number of clockwise edge increases as its horizontal neighbors and the same number of anticlockwise edge increases as its vertical neighbors
%C A206188 Table starts
%C A206188 .......256........1976..........15616.............124048................986388
%C A206188 ......1976.......19688.........205804............2410188..............30417316
%C A206188 .....15616......205804........3802676...........95187756............2674444796
%C A206188 ....124048.....2410188.......95187756.........5674969344..........364197868292
%C A206188 ....986388....30417316.....2674444796.......364197868292........51975660472496
%C A206188 ...7854572...406202048....78597249508.....23825180259760......7481908870312588
%C A206188 ..62623348..5614757320..2345626036136...1565902419680820...1078848431799509828
%C A206188 .499915248.79279051900.70435464744476.103072965502712948.155633082985105845972
%H A206188 R. H. Hardin, <a href="/A206188/b206188.txt">Table of n, a(n) for n = 1..84</a>
%e A206188 Some solutions for n=4 k=3
%e A206188 ..1..0..0..0....3..2..0..0....2..0..2..2....2..3..2..0....0..2..0..3
%e A206188 ..0..3..2..1....2..1..3..1....3..0..0..0....2..3..0..0....0..2..0..3
%e A206188 ..0..2..2..1....0..3..2..2....0..3..2..1....0..0..2..1....2..2..0..3
%e A206188 ..0..1..1..1....3..2..0..3....0..2..2..1....3..0..1..1....0..2..0..3
%e A206188 ..1..0..3..2....1..2..3..3....0..1..1..1....2..2..0..2....2..2..0..3
%Y A206188 Column 1 is A205363
%K A206188 nonn,tabl,new
%O A206188 1,1
%A A206188 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206187
%S A206187 62623348,5614757320,2345626036136,1565902419680820,
%T A206187 1078848431799509828,747343697917253418576
%N A206187 Number of (n+1)X8 0..3 arrays with every 2X2 subblock having the same number of clockwise edge increases as its horizontal neighbors and the same number of anticlockwise edge increases as its vertical neighbors
%C A206187 Column 7 of A206188
%e A206187 Some solutions for n=4
%e A206187 ..1..1..3..1..1..3..1..0....0..1..0..1..1..0..1..0....1..3..2..1..3..0..2..2
%e A206187 ..0..2..1..3..2..1..0..2....0..0..0..1..2..0..0..0....2..2..2..1..3..0..2..1
%e A206187 ..0..0..1..2..2..3..0..1....3..3..0..1..2..3..3..3....3..1..1..1..3..0..2..2
%e A206187 ..3..1..0..1..3..0..1..0....2..1..0..3..3..2..1..0....0..0..0..1..3..0..0..0
%e A206187 ..2..2..1..0..0..2..0..1....1..1..0..3..0..2..1..0....1..3..0..1..3..3..3..0
%K A206187 nonn,new
%O A206187 1,1
%A A206187 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206186
%S A206186 7854572,406202048,78597249508,23825180259760,7481908870312588,
%T A206186 2363371012699739212,747343697917253418576,236405079530904298698860,
%U A206186 74790605792946446088286728,23662388118003270037770957312
%N A206186 Number of (n+1)X7 0..3 arrays with every 2X2 subblock having the same number of clockwise edge increases as its horizontal neighbors and the same number of anticlockwise edge increases as its vertical neighbors
%C A206186 Column 6 of A206188
%H A206186 R. H. Hardin, <a href="/A206186/b206186.txt">Table of n, a(n) for n = 1..23</a>
%e A206186 Some solutions for n=4
%e A206186 ..1..2..0..1..0..0..0....0..1..1..0..0..3..3....0..2..3..2..1..0..3
%e A206186 ..2..2..0..1..0..1..0....0..0..1..1..3..3..0....2..1..1..3..2..1..0
%e A206186 ..1..2..0..0..0..1..0....2..0..1..2..3..0..0....1..3..2..0..3..2..1
%e A206186 ..1..2..2..0..1..1..0....1..0..3..3..2..2..1....2..3..3..0..1..2..3
%e A206186 ..1..1..2..0..0..0..0....0..0..3..0..2..1..1....1..2..0..3..0..1..2
%K A206186 nonn,new
%O A206186 1,1
%A A206186 R. H. Hardin (rhhardin(AT)att.net) Feb 04 2012

%I A206185
%S A206185 986388,30417316,2674444796,364197868292,51975660472496,
%T A206185 7481908870312588,1078848431799509828,155633082985105845972,
%U A206185 22454117874107278681104,3239703783044424175654576,467432771562040406990065172
%N A206185 Number of (n+1)X6 0..3 arrays with every 2X2 subblock having the same number of clockwise edge increases as its horizontal neighbors and the same number of anticlockwise edge increases as its vertical neighbors
%C A206185 Column 5 of A206188
%H A206185 R. H. Hardin, <a href="/A206185/b206185.txt">Table of n, a(n) for n = 1..49</a>
%e A206185 Some solutions for n=4
%e A206185 ..2.