# OEIS Recent Additions (http://oeis.org/recent.txt)
# Last Modified: May 23 07:22 UTC 2012
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%I A212658
%S A212658 1,2,4,8,17,37,86,199,475,1138,2769,6748,16613,40904,101317,251401,
%T A212658 624958,1555940,3882708,9701790,24276866,60817940,152508653,382828565
%N A212658 Number of multisets {1^k1, 2^k2, ..., n^kn}, ki>=0, with the sum of reciprocals <= 1.
%C A212658 The number of distinct sums of reciprocals is given by A212606.
%Y A212658 Cf. A212657, A212606, A212607, A020473, A092669, A092671, A208480.
%K A212658 nonn,more,new
%O A212658 0,2
%A A212658 _Max Alekseyev_, May 23 2012

%I A212657
%S A212657 1,2,3,5,8,14,26,46,83,151,276,503,921,1689,3113,5730,10549,19441,
%T A212657 35868,66209,122316,226157,418373,774394,1434130,2657246,4925837,
%U A212657 9135444,16949660,31460444,58415377,108502932
%N A212657 Number of subsets of {1,2,...,n} with the sum of reciprocals <= 1.
%C A212657 The number of distinct sums of reciprocals is given by A212607.
%Y A212657 Cf. A212658 (reciprocals can appear multiple times)
%Y A212657 Cf. A212606, A212607, A020473, A092669, A092671, A208480.
%K A212657 nonn,more,new
%O A212657 0,2
%A A212657 _Max Alekseyev_, May 23 2012

%I A212580
%S A212580 1,1,2,5,20,102,626,4458,36144
%N A212580 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb where a<b<c.
%C A212580 Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> bac where a<b<c.
%C A212580 Also the number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> cba where a<b<c.
%H A212580 S. Linton, J. Propp, T. Roby, and J. West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%e A212580 Contribution from _Alois P. Heinz_, May 22 2012: (Start)
%e A212580 a(3) = 5: {123, 132}, {213}, {231}, {312}, {321}.
%e A212580 a(4) = 20: {1234, 1243, 1324}, {1342}, {1423}, {1432}, {2134}, {2143}, {2314}, {2341, 2431}, {2413}, {3124}, {3142}, {3214}, {3241}, {3412}, {3421}, {4123, 4132}, {4213}, {4231}, {4312}, {4321}. (End)
%Y A212580 Cf. A210667, A210668, A210669, A210671, A212417, A212581.
%K A212580 nonn,new
%O A212580 0,3
%A A212580 _Tom Roby_, May 21 2012

%I A212581
%S A212581 1,1,2,4,17,89,556,4011,32843,301210
%N A212581 Number of equivalence classes of S_n under transformations of positionally and numerically adjacent elements of the form abc <--> acb <--> bac where a<b<c.
%H A212581 S. Linton, J. Propp, T. Roby, and J. West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%e A212581 Contribution from _Alois P. Heinz_, May 22 2012: (Start)
%e A212581 a(3) = 4: {123, 132, 213}, {231}, {312}, {321}.
%e A212581 a(4) = 17: {1234, 1243, 1324, 2134}, {1342}, {1423}, {1432}, {2143}, {2314}, {2341, 2431, 3241}, {2413}, {3124}, {3142}, {3214}, {3412}, {3421}, {4123, 4132, 4213}, {4231}, {4312}, {4321}. (End)
%Y A212581 Cf. A210667, A210668, A210669, A210671, A212417, A212580.
%K A212581 nonn,new
%O A212581 0,3
%A A212581 _Tom Roby_, May 21 2012
%E A212581 a(9) from _Alois P. Heinz_, May 22 2012

%I A212607
%S A212607 1,2,3,5,8,14,21,38,70,129,238,440,504,949,1790,2301,4363,8272,12408,
%T A212607 23604,26675,45724,87781,168549,181989,351076,677339,1306894,1399054,
%U A212607 2709128,2795144
%N A212607 Number of distinct sums <= 1 of distinct reciprocals of integers <= n.
%e A212607 a(3) = 5 counts numbers { 0, 1/3, 1/2, 5/6, 1 }, each of which is can be represented as the sum of distinct reciprocals 1/1, 1/2, and 1/3.
%p A212607 s:= proc(n) option remember;
%p A212607       `if`(n=0, {0}, map (x->`if`(n*x+1<=n, [x, x+1/n][], x), s(n-1)))
%p A212607     end:
%p A212607 a:= n-> nops(s(n)):
%p A212607 seq (a(n), n=0..20);  # _Alois P. Heinz_, May 22 2012
%Y A212607 For possibly non-distinct reciprocals, see A212606.
%Y A212607 Cf. A020473, A092669, A092671, A208480.
%K A212607 nonn,more,new
%O A212607 0,2
%A A212607 _Max Alekseyev_, May 22 2012
%E A212607 a(27)-a(30) from _Alois P. Heinz_, May 22 2012

%I A212606
%S A212606 1,2,3,6,10,26,34,103,175,393,599,2015,2551,8681,14254,19620,34700,
%T A212606 129557,161272,595304,695175,1094164,1903859,7654850
%N A212606 Number of distinct sums <= 1 of reciprocals of positive integers <= n.
%e A212606 a(3) = 6 counts numbers { 0, 1/3, 1/2, 2/3, 5/6, 1 }, each of which is can be represented as the sum of reciprocals 1/1, 1/2, and 1/3.
%Y A212606 For distinct sums of distinct reciprocals, see A212607.
%Y A212606 Cf. A020473, A092669, A092671, A208480
%K A212606 nonn,more,new
%O A212606 0,2
%A A212606 _Max Alekseyev_, May 22 2012

%I A212600
%S A212600 35,49,55,65,77,85,77,91,119,121,143,161,187,217,247,253,259,289,299,
%T A212600 301,319,323,329,341,343,371,403,427,437,451,473,481,493,497,511,527,
%U A212600 533,539,553,581,589,49,77,91,119,133,143,161,187,209,217,221
%N A212600 Composite numbers p-m! arising in A212266.
%H A212600 Charles R Greathouse IV, <a href="/A212600/b212600.txt">Table of n, a(n) for n = 1..10000</a>
%Y A212600 Cf. A212266.
%K A212600 nonn,new
%O A212600 1,1
%A A212600 _N. J. A. Sloane_, May 22 2012

%I A212598
%S A212598 0,0,1,2,3,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,0,1,2,3,4,5,6,
%T A212598 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,
%U A212598 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66
%N A212598 a(n) = n - m!, where m is the largest number such that m! <= n.
%p A212598 f:=proc(n) local i; for i from 0 to n do if i! > n then break; fi; od; n-(i-1)!; end;
%p A212598 [seq(f(n),n=1..70)];
%Y A212598 Cf. A136437, A212266.
%K A212598 nonn,new
%O A212598 1,4
%A A212598 _N. J. A. Sloane_, May 22 2012

%I A212596
%S A212596 12,192,3456,66048,1296384,25731072,513048576,10248388608,
%T A212596 204867108864,4096536870912,81924294967296,1638434359738368,
%U A212596 32768274877906944,655362199023255552
%N A212596 Number of cards required to build a Menger sponge of level n in origami.
%H A212596 CTRL Byte, 2010-04-13, <a href="http://ctrlbyte.wordpress.com/2010/04/13/2010-04-13-mengersponge/">Menger Sponge Construction</a>
%H A212596 ELJJDX, Choux romanesco, vache qui rit et intégrales curvilignes, <a href="http://eljjdx.canalblog.com/archives/2009/12/20/16221444.html">Am-stram-gram, ticket-ticket-bus-et-tram (French)</a>
%H A212596 Nick Hamblet, Σidiot's Blog, 2009-03-01, <a href="http://sumidiot.wordpress.com/tag/menger-sponge/">Counting Cards</a>
%H A212596 Michel Lucas, <a href="http://www.defi66000.fr/">Défi 66 000 tickets (French)</a>
%H A212596 Dr. Jeannine Mosely, The Institute For Figuring, <a href="http://theiff.org/oexhibits/menger02.html">Business Card Menger Sponge</a>
%H A212596 Nicholas Rougeux, Mengermania, <a href="http://c82.net/mengermania/instructions.php">Instructions</a>
%H A212596 Wikipedia, <a href="http://en.wikipedia.org/wiki/Menger_sponge">Menger sponge</a>
%F A212596 a(n) = 4*(8^n + 2*20^n) = 2^(2*n+3)*5^n+2^(3*n+2)
%e A212596 12 cards (a(0)) are required for a single origami cube : 6 for the cube skeleton, and 6 for panels or possible links to other cubes.
%Y A212596 Equal to A082685*3*4^(n+1)
%K A212596 easy,nonn,new
%O A212596 0,1
%A A212596 _Daniel de Rauglaudre_, May 22 2012

%I A212595
%S A212595 2,2,4,2,2,4,2,4,6,2,2,4,6,2,4,2,2,4,2,4,6,2,4,6,2,2,4,6,2,4,2,2,4,6,
%T A212595 2,4,2,4,6,2,4,6,8,2,4,2,2,4,2,2,4,2,4,6,8,10,12,14,2,4,2,4,6,2,2,4,6,
%U A212595 8,10,2,2,4,6,2,4,6,2,4,2,4,6,2,4,6,2,2
%N A212595 Let f(n) = 2n-7. Difference between f(n) and the nearest prime <  f(n).
%C A212595 It’s known that there is always a prime between n and 2n - 7 for all n > = 10.
%H A212595 Michel Lagneau, <a href="/A212595/b212595.txt">Table of n, a(n) for n = 10..10000</a>
%H A212595 Peter Vandendriessche and Hojoo Lee, <a href="http://www.scribd.com/doc/24487088/Hojoo-Lee-Peter-Vandendriessche-Problems-in-Elementary-Number-Theory">Problems in elementary number theory</a>, Problem E37
%e A212595 a(12) = 4 because 2*12-7 = 17, and the nearest prime p < 17 such that 12 < p < 17 is p = 13. Hence a(12) = 17 - 13 = 4.
%p A212595 with(numtheory):for n from 10 to 100 do:x:=2*n-7:i:=0:for p from x-1 by -1 to n+1 while(i=0) do:if type(p,prime)=true then i:=1:printf(`%d, `,x-p):else fi:od:od:
%K A212595 nonn,new
%O A212595 10,1
%A A212595 _Michel Lagneau_, May 22 2012

%I A212493
%S A212493 0,5,3,3,3,17,13,23,19,19,37,31,31,47,43,59,53,67,61,0,79,73,73,73,73,
%T A212493 0,107,103,127,131,109,113,113,151,113,139,163,157,157,179,173,0,223,
%U A212493 197,193,233,193,191,191,193,199,0,0,257,251,251,0,277,271,271
%N A212493 Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if p<p_n, or the intervals (p_n/2,p/2), (p_n,p], if p>p_n, contain the same number of primes, and a(n)=0, if no such prime p exists.
%C A212493 a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan (A104272) and Labos (A080359) prime (see sequence A164554).
%C A212493 a(n)>p_n if and only if p_n is Labos prime but not Ramanujan prime.
%F A212493 If p_n is not a Labos prime, then a(n)=A080359(n-pi(p_n/2)).
%e A212493 Let n=5, p_5=11; p=2 is not suitable, since in (1,5.5) we have 3 primes, while in (2,11] there are 4 primes. Consider p=3. Now in intervals (1.5,5.5) and (3,11] we have the same number (3) of primes. Therefore, a(5)=3. The same value we can obtain by the formula. Since 11 is not a labos prime, then a(5)=A080359(5-pi(5.5))=A080359(2)=3.
%Y A212493 Cf. A104272, A080359, A164554.
%K A212493 nonn,new
%O A212493 1,2
%A A212493 _Vladimir Shevelev_ and Peter Moses, May 18 2012

%I A212541
%S A212541 0,11,11,11,7,17,13,29,29,23,41,41,37,47,43,59,53,67,61,0,97,97,97,97,
%T A212541 89,0,107,103,127,149,109,149,149,151,137,139,167,167,163,179,173,0,
%U A212541 227,229,229,233,229,227,223,211,199,0,0,263,263,257,0,281,281
%N A212541 Let p_n=prime(n), n>=1. Then a(n) is the maximal prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if p<p_n, or the intervals (p_n/2,p/2), (p_n,p], if p>p_n, contain the same number of primes, and a(n)=0, if no such prime p exists.
%C A212541 a(n)<p_n if and only if p_n is Ramanujan prime (A104272).
%C A212541 a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan and Labos (A080359) prime (see sequence A164554).
%F A212541 If p_n is not a Ramanujan prime, then a(n) = A104272(n-pi(p_n/2)).
%e A212541 Let n=4, p_n=7. Since 7 is not Ramanujan prime, then a(4) = A104272(4-pi(3.5)) = A104272(2) = 11.
%Y A212541 Cf. A212493, A104272, A080359, A164554.
%K A212541 nonn,new
%O A212541 1,2
%A A212541 _Vladimir Shevelev_ and Peter Moses, May 20 2012

%I A212592
%S A212592 1,6,11,106,201,2022,3843,38794,73745,744646,1415547,14293930,
%T A212592 27172313,274381478,521590643,5266936010,10012281377,101102361990,
%U A212592 192192442603,1940727511786,3689262580969,37253563629926,70817864678883,715107089849866
%N A212592 a(n) is the difference between multiples of 7 with even and odd digit sum in base 6 in interval [0,6^n).
%H A212592 Vladimir Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177v2 [math.NT], 2007
%H A212592 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,21,0,-35,0,7).
%F A212592 For n>=7, a(n) = 21*a(n-2)-35*a(n-4)+7*a(n-6).
%F A212592 G.f.: x*(1+6*x-10*x^2-20*x^3+5*x^4+6*x^5)/(1-21*x^2+35*x^4-7*x^6). [_Bruno Berselli_, May 22 2012]
%t A212592 LinearRecurrence[{0, 21, 0, -35, 0, 7}, {1, 6, 11, 106, 201, 2022}, 24] (* _Bruno Berselli_, May 22 2012 *)
%Y A212592 Cf. A038754, A212500, A091042.
%K A212592 nonn,base,easy,new
%O A212592 1,2
%A A212592 _Vladimir Shevelev_ and Peter Moses, May 22 2012

%I A212593
%S A212593 1,8,15,232,449,7400,14351,237832,461313,7648968,14836623,246015528,
%T A212593 477194433,7912700328,15348206223,254499628104,493651049985,
%U A212593 8185582834056,15877514618127,263276481572712,510675448527297,8467876653984360
%N A212593 a(n) is the difference between multiples of 9 with even and odd digit sum in base 8 in interval [0,8^n).
%H A212593 Vladimir Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177v2 [math.NT], 2007
%H A212593 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,36,0,-126,0,84,0,-9).
%F A212593 For n>=9, a(n) = 36*a(n-2)-126*a(n-4)+84*a(n-6)-9*a(n-8).
%F A212593 G.f.: x*(1+8*x-21*x^2-56*x^3+35*x^4+56*x^5-7*x^6-8*x^7)/((1-3*x^2)*(1-33*x^2+27*x^4-3*x^6)). [_Bruno Berselli_, May 22 2012]
%t A212593 LinearRecurrence[{0, 36, 0, -126, 0, 84, 0, -9}, {1, 8, 15, 232, 449, 7400, 14351, 237832}, 22] (* _Bruno Berselli_, May 22 2012 *)
%Y A212593 Cf. A038754, A212500, A212592, A091042.
%K A212593 nonn,base,easy,new
%O A212593 1,2
%A A212593 _Vladimir Shevelev_ and Peter Moses, May 22 2012

%I A212594
%S A212594 1,10,19,430,841,20602,40363,995710,1951057,48162410,94373763,
%T A212594 2329795534,4565217305,112701782490,220838347675,5451852478622,
%U A212594 10682866609569,263728727794378,516774588979187,12757653047779310,24998531506579433,617140623134480698
%N A212594 a(n) is the difference between multiples of 11 with even and odd decimal digit sum in interval [0,10^n).
%H A212594 Vladimir Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>, arXiv:0710.3177v2 [math.NT], 2007
%H A212594 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (0,55,0,-330,0,462,0,-165,0,11).
%F A212594 For n>=11, a(n) = 55*a(n-2)-330*a(n-4)+462*a(n-6)-165*a(n-8)+11*a(n-10).
%F A212594 G.f.: x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10). [_Bruno Berselli_, May 22 2012]
%t A212594 LinearRecurrence[{0, 55, 0, -330, 0, 462, 0, -165, 0, 11}, {1, 10, 19, 430, 841, 20602, 40363, 995710, 1951057, 48162410}, 22] (* _Bruno Berselli_, May 22 2012 *)
%o A212594 (MAGMA) m:=23; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+10*x-36*x^2-120*x^3+126*x^4+252*x^5-84*x^6-120*x^7+9*x^8+10*x^9)/(1-55*x^2+330*x^4-462*x^6+165*x^8-11*x^10))); // _Bruno Berselli_, May 22 2012
%Y A212594 Cf. A038754, A212500, A212592, A212593, A091042.
%K A212594 nonn,base,easy,new
%O A212594 1,2
%A A212594 _Vladimir Shevelev_ and Peter Moses, May 22 2012

%I A212583
%S A212583 66161,534851,3152573
%N A212583 Primes p such that p^2 divides 6^(p-1) - 1.
%C A212583 Base 6 Wieferich primes.
%D A212583 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996, page 347
%H A212583 Richard Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sort">Thema: Fermatquotient B^P-1 == 1 mod (P^2)</a>
%H A212583 François G. Dorais and Dominic Klyve, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Klyve/klyve3.pdf">A Wieferich prime search up to p < 6.7*10^15</a>, J. Integer Seq. 14 (2011), Art. 11.9.2, 1-14.
%H A212583 Wilfrid Keller and Jörg Richstein, <a href="http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html">Fermat quotients q_p(a) that are divisible by p</a>.
%H A212583 Eric Weisstein, <a href="http://mathworld.wolfram.com/FermatQuotient.html">Fermat Quotient</a>, MathWorld
%H A212583 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wieferich_prime#Base-a_Wieferich_primes">Base-a Wieferich primes</a>
%Y A212583 Cf. A001220, A014127, A090968, A123692, A123693, A128667, A128668, A128669.
%K A212583 nonn,hard,bref,more,new
%O A212583 1,1
%A A212583 _Felix Fröhlich_, May 22 2012

%I A212001
%S A212001 1,4,3,9,8,5,20,19,16,11,35,34,31,26,15,66,65,62,57,46,31,105,104,101,
%T A212001 96,85,70,39,176,175,172,167,156,141,110,71,270,269,266,261,250,235,
%U A212001 204,165,94,420,419,416,411,400,385,354,315,244,150,616,615
%N A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.
%C A212001 The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.
%F A212001 T(n,k) = A066186(n) - A066186(k-1).
%F A212001 T(n,k) = Sum_{j=k..n} A138879(j).
%e A212001 For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below the sum of all parts of each set:
%e A212001 --------------------------------------------------------
%e A212001 .  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
%e A212001 --------------------------------------------------------
%e A212001 .  The        Last       Last       Last       The
%e A212001 .  five       four       three      two        last
%e A212001 .  shells     shells     shells     shells     shell
%e A212001 .  of 5       of 5       of 5       of 5       of 5
%e A212001 --------------------------------------------------------
%e A212001 .
%e A212001 .  5          5          5          5          5
%e A212001 .  3+2        3+2        3+2        3+2        3+2
%e A212001 .  4+1        4+1        4+1        4+1          1
%e A212001 .  2+2+1      2+2+1      2+2+1      2+2+1          1
%e A212001 .  3+1+1      3+1+1      3+1+1        1+1          1
%e A212001 .  2+1+1+1    2+1+1+1      1+1+1        1+1          1
%e A212001 .  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
%e A212001 . ---------- ---------- ---------- ---------- ----------
%e A212001 .     35         34         31         26         15
%e A212001 .
%e A212001 So row 5 lists 35, 34, 31, 26, 15.
%e A212001 .
%e A212001 Triangle begins:
%e A212001 1;
%e A212001 4,     3;
%e A212001 9,     8,   5;
%e A212001 20,   19,  16,  11;
%e A212001 35,   34,  31,  26,  15;
%e A212001 66,   65,  62,  57,  46,  31;
%e A212001 105, 104, 101,  96,  85,  70,  39;
%e A212001 176, 175, 172, 167, 156, 141, 110,  71;
%e A212001 270, 269, 266, 261, 250, 235, 204, 165,  94;
%e A212001 420, 419, 416, 411, 400, 385, 354, 315, 244, 150;
%Y A212001 Mirror of triangle A212011. Column 1 is A066186. Right border is A138879.
%Y A212001 Cf. A135010, A138121, A182703, A206563, A211980, A212000, A212010.
%K A212001 nonn,tabl,new
%O A212001 1,2
%A A212001 _Omar E. Pol_, Apr 26 2012

%I A211980
%S A211980 1,2,1,3,2,1,5,4,3,2,7,6,5,4,2,11,10,9,8,6,4,15,14,13,12,10,8,4,22,21,
%T A211980 20,19,17,15,11,7,30,29,28,27,25,23,19,15,8,42,41,40,39,37,35,31,27,
%U A211980 20,12,56,55,54,53,51,49,45,41,34,26,14,77,76,75
%N A211980 Triangle read by rows: T(n,k) = total number of regions in the last n-k+1 shells of n.
%C A211980 The set of partitions of n contains n shells and A000041(n) regions. For the definition of "outer shell of n" see A135010. For the definition of "region of n" see A206437.
%H A211980 O. E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%F A211980 T(n,1) = A000041(n).
%F A211980 T(n,k) = A000041(n) - A000041(k-1), 1<k<=n.
%F A211980 T(n,k) = Sum_{j=k..n} A187219(j).
%e A211980 Triangle begins:
%e A211980 1;
%e A211980 2,   1;
%e A211980 3,   2,  1;
%e A211980 5,   4,  3,  2;
%e A211980 7,   6,  5,  4,  2;
%e A211980 11, 10,  9,  8,  6,  4;
%e A211980 15, 14, 13, 12, 10,  8,  4;
%e A211980 22, 21, 20, 19, 17, 15, 11,  7;
%e A211980 30, 29, 28, 27, 25, 23, 19, 15,  8;
%e A211980 42, 41, 40, 39, 37, 35, 31, 27, 20, 12;
%e A211980 56, 55, 54, 53, 51, 49, 45, 41, 34, 26, 14;
%e A211980 77, 76, 75, 74, 72, 70, 66, 62, 55, 47, 35, 21;
%Y A211980 Mirror of triangle A211990. Column 1 is A000041, n >= 1. Right border is A187219.
%Y A211980 Cf. A135010, A138121, A182703, A186114, A206437, A212000, A212001.
%K A211980 nonn,tabl,new
%O A211980 1,2
%A A211980 _Omar E. Pol_, Apr 27 2012

%I A212011
%S A212011 1,3,4,5,8,9,11,16,19,20,15,26,31,34,35,31,46,57,62,65,66,39,70,85,96,
%T A212011 101,104,105,71,110,141,156,167,172,175,176,94,165,204,235,250,261,
%U A212011 266,269,270,150,244,315,354,385,400,411,416,419,420,196,346
%N A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.
%C A212011 The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.
%F A212011 T(n,k) = A066186(n) - A066186(n-k).
%F A212011 T(n,k) = Sum_{j=n-k+1..n} A138879(j).
%e A212011 For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of in each set:
%e A212011 --------------------------------------------------------
%e A212011 .  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
%e A212011 --------------------------------------------------------
%e A212011 .  The        Last       Last       Last       The
%e A212011 .  last       two        three      four       five
%e A212011 .  shell      shells     shells     shells     shells
%e A212011 .  of 5       of 5       of 5       of 5       of 5
%e A212011 --------------------------------------------------------
%e A212011 .
%e A212011 .  5          5          5          5          5
%e A212011 .  3+2        3+2        3+2        3+2        3+2
%e A212011 .    1        4+1        4+1        4+1        4+1
%e A212011 .      1      2+2+1      2+2+1      2+2+1      2+2+1
%e A212011 .      1        1+1      3+1+1      3+1+1      3+1+1
%e A212011 .        1        1+1      1+1+1    2+1+1+1    2+1+1+1
%e A212011 .          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
%e A212011 . ---------- ---------- ---------- ---------- ----------
%e A212011 .     15         26         31         34         35
%e A212011 .
%e A212011 So row 5 lists 15, 26, 31, 34, 35.
%e A212011 .
%e A212011 Triangle begins:
%e A212011 1;
%e A212011 3,     4;
%e A212011 5,     8,   9;
%e A212011 11,   16,  19,  20;
%e A212011 15,   26,  31,  34,  35;
%e A212011 31,   46,  57,  62,  65,  66;
%e A212011 39,   70,  85,  96, 101, 104, 105;
%e A212011 71,  110, 141, 156, 167, 172, 175, 176;
%e A212011 94,  165, 204, 235, 250, 261, 266, 269, 270;
%e A212011 150, 244, 315, 354, 385, 400, 411, 416, 419, 420;
%Y A212011 Mirror of triangle A212001. Column 1 is A138879. Right border is A066186.
%Y A212011 Cf. A135010, A182703, A212000, A212010.
%K A212011 nonn,tabl,new
%O A212011 1,2
%A A212011 _Omar E. Pol_, Apr 26 2012

%I A212502
%S A212502 4,8,12,16,24,32,36,48,56,64,72,96,108,112,128,132,143,144,156,168,
%T A212502 192,216,224,256,264,272,288,312,324,336,384,392,396,399,432,448,468,
%U A212502 496,504,512,527,528,544,552,576,624,648,672,768,779,784,792,816,864
%N A212502 2-Gaussian pseudoprimes
%C A212502 We say that an integer n is t-Gaussian pseudoprime if 1+t^2 is invertible modulo n and ((1-t^2)/(1+t^2)+ i*2t/(1+t^2))^A201629(n) == 1 (mod n).
%t A212502 A201629[n_]:=Which[Mod[n,4]==3,n+1,Mod[n,4]==1,n-1,True,n];Inv[a_,mod_]:=If[GCD[a,mod]>1,0,Last@Reduce[a*x==1,x,Modulus->mod]];zz[t_,mod_]:=(1-t^2)*Inv[1+t^2,mod]+I*2 t*Inv[1+t^2,mod];test[t_,M_]:=PowerMod[zz[t,M],A201629[M],M]===1;Select[1+Range[1000],!PrimeQ[#]&&test[2,#]&]
%K A212502 nonn,new
%O A212502 1,1
%A A212502 _José María Grau Ribas_, May 19 2012

%I A212138
%S A212138 1,2,0,3,0,1,4,0,2,0,5,0,4,0,1,6,0,6,2,2,0,7,0,9,4,5,0,1,8,0,12,8,10,
%T A212138 2,2,0,9,0,16,14,18,8,6,0,1,10,0,20,22,32,20,14,4,2,0,11,0,25,32,52,
%U A212138 42,34,14,7,0,1,12,0,30,46,80,80,72,42,22,4,2,0,13,0,36,62,119,1
%N A212138 Triangular array:  T(n,k) is the number of k-element subsets S of {1,...,n} whose average is in S.
%e A212138 First 7 rows:
%e A212138 1
%e A212138 2...0
%e A212138 3...0...1
%e A212138 4...0...2...0
%e A212138 5...0...4...0...1
%e A212138 6...0...6...2...2...0
%e A212138 7...0...9...4...5...0...1
%e A212138 T(5,3) counts these subsets: {1,2,3}, 1,3,5}, {2,3,4}, {3,4,5}.
%t A212138 t[n_, k_] := Length[Flatten[Map[Apply[Intersection, #] &,
%t A212138     Select[Map[{#, {Mean[#]}} &, Subsets[Range[n], {k}]], IntegerQ[Last[Last[#]]] &]]]]
%t A212138 Flatten[Table[t[n, k], {n, 1, 12}, {k, 1, n}]]
%t A212138 TableForm[Table[t[n, k], {n, 1, 12}, {k, 1, n}]]
%t A212138 (* Peter Moses, May 1 2012 *)
%Y A212138 Cf. A061865.
%K A212138 nonn,new
%O A212138 1,2
%A A212138 _Clark Kimberling_, May 06 2012

%I A212137
%S A212137 0,0,1,0,2,0,0,4,2,1,0,6,6,4,0,0,9,12,11,4,1,0,12,22,26,16,6,0,0,16,
%T A212137 36,52,44,24,6,1,0,20,54,94,100,70,30,8,0,0,25,78,156,200,176,102,39,
%U A212137 8,1,0,30,108,246,368,386,282,144,48,10,0,0,36,144,369,632,772
%N A212137 Triangular array:  T(n,k) is the number of k-element subsets of {1,...,n} whose average is not an integer.
%C A212137 Alternating row sums:  -1,-2,-3,-4,-5,-6,...
%C A212137 Let S(n,k) be the number in row n and column k of the array A061865;  then S(n,k)+T(n,k)=C(n,k), for 1<=k<=n, n>=1.
%e A212137 First 7 rows:
%e A212137 0
%e A212137 0...1
%e A212137 0...2....0
%e A212137 0...4....2....1
%e A212137 0...6....6....4....0
%e A212137 0...9....12...11...4....1
%e A212137 0...12...22...26...16...6...0
%t A212137 t[n_, k_] := t[n, k] =
%t A212137   Count[Map[IntegerQ[Mean[#]] &, Subsets[Range[n], {k}]], False]
%t A212137 Flatten[Table[t[n, k], {n, 1, 12}, {k, 1, n}]]
%t A212137 TableForm[Table[t[n, k], {n, 1, 12}, {k, 1, n}]]
%t A212137 s[n_] := Sum[t[n, k], {k, 1, n}]
%t A212137 (* Peter Moses, May 1 2012 *)
%Y A212137 Cf. A061865.
%K A212137 nonn,new
%O A212137 1,5
%A A212137 _Clark Kimberling_, May 06 2012

%I A212135
%S A212135 0,0,4,24,84,220,480,924,1624,2664,4140,6160,8844,12324,16744,22260,
%T A212135 29040,37264,47124,58824,72580,88620,107184,128524,152904,180600,
%U A212135 211900,247104,286524,330484,379320,433380,493024,558624,630564
%N A212135 Number of (w,x,y,z) with all terms in {1,...,n} and median<mean.
%C A212135 Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>mean.  A212135(n)+A212134(n)=n^4.  For a guide to related sequences, see A211795.
%F A212135 a(n)=n*(n - 1)*(n^2 - n + 2)/2.
%F A212135 a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).
%t A212135 t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[
%t A212135 Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 > (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212135 Flatten[Map[{t[#]} &, Range[0, 20]]]  (* A212135 *)
%t A212135 %/4 (* A002817 *)
%Y A212135 Cf. A211795.
%K A212135 nonn,new
%O A212135 0,3
%A A212135 _Clark Kimberling_, May 05 2012

%I A212134
%S A212134 0,1,12,57,172,405,816,1477,2472,3897,5860,8481,11892,16237,21672,
%T A212134 28365,36496,46257,57852,71497,87420,105861,127072,151317,178872,
%U A212134 210025,245076,284337,328132,376797,430680,490141,555552,627297,705772
%N A212134 Number of (w,x,y,z) with all terms in {1,...,n} and median<=mean.
%C A212134 Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>=mean.   A212134(n)+ A212135(n)=n^4.  For a guide to related sequences, see A211795.
%F A212134 a(n)=n(n^3+2n^2-3n+2)/2.
%F A212134 a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).
%t A212134 t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 <= (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #},
%t A212134 {z, 1, #}] &[n]; s)]];
%t A212134 Flatten[Map[{t[#]} &, Range[0, 50]]]  (* A212134 *)
%t A212134 (* Peter Moses, May 1 2012 *)
%Y A212134 Cf. A211795, A212133.
%K A212134 nonn,new
%O A212134 0,3
%A A212134 _Clark Kimberling_, May 04 2012

%I A212133
%S A212133 0,1,8,33,88,185,336,553,848,1233,1720,2321,3048,3913,4928,6105,7456,
%T A212133 8993,10728,12673,14840,17241,19888,22793,25968,29425,33176,37233,
%U A212133 41608,46313,51360,56761,62528,68673,75208,82145,89496,97273
%N A212133 Number of (w,x,y,z) with all terms in {1,...,n} and median=mean.
%C A212133 For a guide to related sequences, see A211795.
%F A212133 a(n)=4a(n-1)-6a(n-2)+4a(n-3)-a(n-4).
%e A212133 a(2) counts these 4-tuples:  (1,1,1,1), (1,1,2,2), (1,2,1,2), (2,1,1,2), (1,2,2,1), (2,1,2,1), (2,2,1,1), (2,2,2,2).
%t A212133 t = Compile[{{n, _Integer}},
%t A212133 Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 == (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212133 Flatten[Map[{t[#]} &, Range[0, 50]]] (* A212133 *)
%t A212133 (* Peter Moses, May 1 2012 *)
%Y A212133 Cf. A211795.
%K A212133 nonn,new
%O A212133 0,3
%A A212133 _Clark Kimberling_, May 04 2012

%I A212543
%S A212543 0,0,1,1,3,4,8,10,17,22,33,42,60,75,103,128,169,209,271,331,421,513,
%T A212543 642,777,963,1158,1421,1703,2070,2471,2985,3546,4257,5043,6019,7105,
%U A212543 8443,9933,11752,13790,16247,19012,22326,26052,30492,35500,41420,48108,55980
%N A212543 Number of partitions of n containing at least one part m-3 if m is the largest part.
%H A212543 Alois P. Heinz, <a href="/A212543/b212543.txt">Table of n, a(n) for n = 3..1000</a>
%F A212543 G.f.: Sum_{i>0} x^(2*i+3) / Product_{j=1..3+i} (1-x^j).
%e A212543 a(5) = 1: [4,1].
%e A212543 a(6) = 1: [4,1,1].
%e A212543 a(7) = 3: [4,1,1,1], [4,2,1], [5,2].
%e A212543 a(8) = 4: [4,1,1,1,1], [4,2,1,1], [4,3,1], [5,2,1].
%e A212543 a(9) = 8: [4,1,1,1,1,1], [4,2,1,1,1], [4,2,2,1], [4,3,1,1], [4,4,1], [5,2,1,1], [5,2,2], [6,3].
%p A212543 b:= proc(n, i) option remember;
%p A212543       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212543     end:
%p A212543 a:= n-> add (b(n-2*m-3, min(n-2*m-3, m+3)), m=1..(n-3)/2):
%p A212543 seq (a(n), n=3..60);
%Y A212543 Column k=3 of A212551.
%K A212543 nonn,new
%O A212543 3,5
%A A212543 _Alois P. Heinz_, May 20 2012

%I A212544
%S A212544 0,0,1,1,3,4,8,11,18,24,37,48,69,89,122,155,207,259,337,419,534,657,
%T A212544 827,1008,1252,1518,1864,2246,2736,3276,3960,4722,5668,6727,8032,9492,
%U A212544 11274,13279,15696,18424,21694,25380,29772,34736,40604,47244,55060,63897
%N A212544 Number of partitions of n containing at least one part m-4 if m is the largest part.
%H A212544 Alois P. Heinz, <a href="/A212544/b212544.txt">Table of n, a(n) for n = 4..1000</a>
%F A212544 G.f.: Sum_{i>0} x^(2*i+4) / Product_{j=1..4+i} (1-x^j).
%e A212544 a(6) = 1: [5,1].
%e A212544 a(7) = 1: [5,1,1].
%e A212544 a(8) = 3: [5,1,1,1], [5,2,1], [6,2].
%e A212544 a(9) = 4: [5,1,1,1,1], [5,2,1,1], [5,3,1], [6,2,1].
%e A212544 a(10) = 8: [5,1,1,1,1,1], [5,2,1,1,1], [5,2,2,1], [5,3,1,1], [5,4,1], [6,2,1,1], [6,2,2], [7,3].
%p A212544 b:= proc(n, i) option remember;
%p A212544       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212544     end:
%p A212544 a:= n-> add (b(n-2*m-4, min(n-2*m-4, m+4)), m=1..(n-4)/2):
%p A212544 seq (a(n), n=4..60);
%Y A212544 Column k=4 of A212551.
%K A212544 nonn,new
%O A212544 4,5
%A A212544 _Alois P. Heinz_, May 20 2012

%I A212545
%S A212545 0,0,1,1,3,4,8,11,19,25,39,52,75,98,137,175,236,300,393,493,635,787,
%T A212545 997,1227,1531,1869,2309,2796,3420,4119,4994,5979,7201,8574,10260,
%U A212545 12164,14470,17082,20225,23778,28025,32838,38542,45011,52642,61286,71434,82937
%N A212545 Number of partitions of n containing at least one part m-5 if m is the largest part.
%H A212545 Alois P. Heinz, <a href="/A212545/b212545.txt">Table of n, a(n) for n = 5..1000</a>
%F A212545 G.f.: Sum_{i>0} x^(2*i+5) / Product_{j=1..5+i} (1-x^j).
%e A212545 a(7) = 1: [6,1].
%e A212545 a(8) = 1: [6,1,1].
%e A212545 a(9) = 3: [6,1,1,1], [6,2,1], [7,2].
%e A212545 a(10) = 4: [6,1,1,1,1], [6,2,1,1], [6,3,1], [7,2,1].
%e A212545 a(11) = 8: [6,1,1,1,1,1], [6,2,1,1,1], [6,2,2,1], [6,3,1,1], [6,4,1], [7,2,1,1], [7,2,2], [8,3].
%p A212545 b:= proc(n, i) option remember;
%p A212545       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212545     end:
%p A212545 a:= n-> add (b(n-2*m-5, min(n-2*m-5, m+5)), m=1..(n-5)/2):
%p A212545 seq (a(n), n=5..60);
%Y A212545 Column k=5 of A212551.
%K A212545 nonn,new
%O A212545 5,5
%A A212545 _Alois P. Heinz_, May 20 2012

%I A212546
%S A212546 0,0,1,1,3,4,8,11,19,26,40,54,79,104,146,190,257,330,436,552,715,896,
%T A212546 1140,1415,1777,2184,2711,3308,4063,4922,5995,7214,8720,10435,12525,
%U A212546 14910,17793,21076,25016,29507,34850,40941,48148,56351,66007,76995,89855,104484
%N A212546 Number of partitions of n containing at least one part m-6 if m is the largest part.
%H A212546 Alois P. Heinz, <a href="/A212546/b212546.txt">Table of n, a(n) for n = 6..1000</a>
%F A212546 G.f.: Sum_{i>0} x^(2*i+6) / Product_{j=1..6+i} (1-x^j).
%e A212546 a(8) = 1: [7,1].
%e A212546 a(9) = 1: [7,1,1].
%e A212546 a(10) = 3: [7,1,1,1], [7,2,1], [8,2].
%e A212546 a(11) = 4: [7,1,1,1,1], [7,2,1,1], [7,3,1], [8,2,1].
%e A212546 a(12) = 8: [7,1,1,1,1,1], [7,2,1,1,1], [7,2,2,1], [7,3,1,1], [7,4,1], [8,2,1,1], [8,2,2], [9,3].
%p A212546 b:= proc(n, i) option remember;
%p A212546       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212546     end:
%p A212546 a:= n-> add (b(n-2*m-6, min(n-2*m-6, m+6)), m=1..(n-6)/2):
%p A212546 seq (a(n), n=6..60);
%Y A212546 Column k=6 of A212551.
%K A212546 nonn,new
%O A212546 6,5
%A A212546 _Alois P. Heinz_, May 20 2012

%I A212547
%S A212547 0,0,1,1,3,4,8,11,19,26,41,55,81,108,152,199,272,351,467,596,776,979,
%T A212547 1255,1566,1978,2448,3054,3747,4628,5635,6896,8342,10125,12172,14673,
%U A212547 17537,21005,24981,29748,35210,41718,49161,57974,68049,79902,93440,109295
%N A212547 Number of partitions of n containing at least one part m-7 if m is the largest part.
%H A212547 Alois P. Heinz, <a href="/A212547/b212547.txt">Table of n, a(n) for n = 7..1000</a>
%F A212547 G.f.: Sum_{i>0} x^(2*i+7) / Product_{j=1..7+i} (1-x^j).
%e A212547 a(9) = 1: [8,1].
%e A212547 a(10) = 1: [8,1,1].
%e A212547 a(11) = 3: [8,1,1,1], [8,2,1], [9,2].
%e A212547 a(12) = 4: [8,1,1,1,1], [8,2,1,1], [8,3,1], [9,2,1].
%e A212547 a(13) = 8: [8,1,1,1,1,1], [8,2,1,1,1], [8,2,2,1], [8,3,1,1], [8,4,1], [9,2,1,1], [9,2,2], [10,3].
%p A212547 b:= proc(n, i) option remember;
%p A212547       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212547     end:
%p A212547 a:= n-> add (b(n-2*m-7, min(n-2*m-7, m+7)), m=1..(n-7)/2):
%p A212547 seq (a(n), n=7..60);
%Y A212547 Column k=7 of A212551.
%K A212547 nonn,new
%O A212547 7,5
%A A212547 _Alois P. Heinz_, May 20 2012

%I A212548
%S A212548 0,0,1,1,3,4,8,11,19,26,41,56,82,110,156,205,281,366,488,627,821,1041,
%T A212548 1340,1684,2135,2657,3331,4108,5095,6238,7663,9315,11354,13709,16588,
%U A212548 19915,23936,28580,34154,40573,48225,57031,67452,79428,93530,109695,128639
%N A212548 Number of partitions of n containing at least one part m-8 if m is the largest part.
%H A212548 Alois P. Heinz, <a href="/A212548/b212548.txt">Table of n, a(n) for n = 8..1000</a>
%F A212548 G.f.: Sum_{i>0} x^(2*i+8) / Product_{j=1..8+i} (1-x^j).
%e A212548 a(10) = 1: [9,1].
%e A212548 a(11) = 1: [9,1,1].
%e A212548 a(12) = 3: [9,1,1,1], [9,2,1], [10,2].
%e A212548 a(13) = 4: [9,1,1,1,1], [9,2,1,1], [9,3,1], [10,2,1].
%e A212548 a(14) = 8: [9,1,1,1,1,1], [9,2,1,1,1], [9,2,2,1], [9,3,1,1], [9,4,1], [10,2,1,1], [10,2,2], [11,3].
%p A212548 b:= proc(n, i) option remember;
%p A212548       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212548     end:
%p A212548 a:= n-> add (b(n-2*m-8, min(n-2*m-8, m+8)), m=1..(n-8)/2):
%p A212548 seq (a(n), n=8..60);
%Y A212548 Column k=8 of A212551.
%K A212548 nonn,new
%O A212548 8,5
%A A212548 _Alois P. Heinz_, May 20 2012

%I A212549
%S A212549 0,0,1,1,3,4,8,11,19,26,41,56,83,111,158,209,287,375,503,648,852,1086,
%T A212549 1403,1770,2255,2817,3546,4393,5469,6723,8294,10120,12382,15011,18228,
%U A212549 21965,26497,31749,38069,45383,54114,64204,76176,89975,106259,124998,146987
%N A212549 Number of partitions of n containing at least one part m-9 if m is the largest part.
%H A212549 Alois P. Heinz, <a href="/A212549/b212549.txt">Table of n, a(n) for n = 9..1000</a>
%F A212549 G.f.: Sum_{i>0} x^(2*i+9) / Product_{j=1..9+i} (1-x^j).
%e A212549 a(11) = 1: [10,1].
%e A212549 a(12) = 1: [10,1,1].
%e A212549 a(13) = 3: [10,1,1,1], [10,2,1], [11,2].
%e A212549 a(14) = 4: [10,1,1,1,1], [10,2,1,1], [10,3,1], [11,2,1].
%e A212549 a(15) = 8: [10,1,1,1,1,1], [10,2,1,1,1], [10,2,2,1], [10,3,1,1], [10,4,1], [11,2,1,1], [11,2,2], [12,3].
%p A212549 b:= proc(n, i) option remember;
%p A212549       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212549     end:
%p A212549 a:= n-> add (b(n-2*m-9, min(n-2*m-9, m+9)), m=1..(n-9)/2):
%p A212549 seq (a(n), n=9..60);
%Y A212549 Column k=9 of A212551.
%K A212549 nonn,new
%O A212549 9,5
%A A212549 _Alois P. Heinz_, May 20 2012

%I A212550
%S A212550 0,0,1,1,3,4,8,11,19,26,41,56,83,112,159,211,291,381,512,663,873,1117,
%T A212550 1448,1833,2342,2938,3708,4611,5760,7105,8792,10769,13215,16077,19585,
%U A212550 23679,28651,34447,41424,49541,59248,70509,83892,99390,117695,138846,163708
%N A212550 Number of partitions of n containing at least one part m-10 if m is the largest part.
%H A212550 Alois P. Heinz, <a href="/A212550/b212550.txt">Table of n, a(n) for n = 10..1000</a>
%F A212550 G.f.: Sum_{i>0} x^(2*i+10) / Product_{j=1..10+i} (1-x^j).
%e A212550 a(12) = 1: [11,1].
%e A212550 a(13) = 1: [11,1,1].
%e A212550 a(14) = 3: [11,1,1,1], [11,2,1], [12,2].
%e A212550 a(15) = 4: [11,1,1,1,1], [11,2,1,1], [11,3,1], [12,2,1].
%e A212550 a(16) = 8: [11,1,1,1,1,1], [11,2,1,1,1], [11,2,2,1], [11,3,1,1], [11,4,1], [12,2,1,1], [12,2,2], [13,3].
%p A212550 b:= proc(n, i) option remember;
%p A212550       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212550     end:
%p A212550 a:= n-> add (b(n-2*m-10, min(n-2*m-10, m+10)), m=1..(n-10)/2):
%p A212550 seq (a(n), n=10..60);
%Y A212550 Column k=10 of A212551.
%K A212550 nonn,new
%O A212550 10,5
%A A212550 _Alois P. Heinz_, May 20 2012

%I A212551
%S A212551 1,0,0,1,0,0,1,1,0,0,2,1,1,0,0,2,3,1,1,0,0,4,3,3,1,1,0,0,4,6,4,3,1,1,
%T A212551 0,0,7,7,7,4,3,1,1,0,0,8,11,9,8,4,3,1,1,0,0,12,13,15,10,8,4,3,1,1,0,0,
%U A212551 14,20,18,17,11,8,4,3,1,1,0,0
%N A212551 Number of partitions T(n,k) of n containing at least one other part m-k if m is the largest part; triangle T(n,k), n>=0, 0<=k<=n.
%C A212551 Reversed rows converge to A024786.
%H A212551 Alois P. Heinz, <a href="/A212551/b212551.txt">Rows n = 0..140, flattened</a>
%F A212551 G.f. of column k: delta_{0,k} + Sum_{i>0} x^(2*i+k) / Product_{j=1..k+i} (1-x^j), where delta is the Kronecker symbol.
%e A212551 T(4,0) = 2: [1,1,1,1], [2,2].
%e A212551 T(4,1) = 1: [2,1,1].
%e A212551 T(5,1) = 3: [2,1,1,1], [2,2,1], [3,2].
%e A212551 T(6,2) = 3: [3,1,1,1], [3,2,1], [4,2].
%e A212551 T(7,2) = 4: [3,1,1,1,1], [3,2,1,1], [3,3,1], [4,2,1].
%e A212551 T(8,4) = 3: [5,1,1,1], [5,2,1], [6,2].
%e A212551 Triangle T(n,k) begins:
%e A212551 1;
%e A212551 0, 0;
%e A212551 1, 0, 0;
%e A212551 1, 1, 0, 0;
%e A212551 2, 1, 1, 0, 0;
%e A212551 2, 3, 1, 1, 0, 0;
%e A212551 4, 3, 3, 1, 1, 0, 0;
%e A212551 4, 6, 4, 3, 1, 1, 0, 0;
%e A212551 7, 7, 7, 4, 3, 1, 1, 0, 0;
%p A212551 b:= proc(n, i) option remember;
%p A212551       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))
%p A212551     end:
%p A212551 T:= (n, k)-> `if`(n=0 and k=0, 1,
%p A212551     add (b(n-2*m-k, min(n-2*m-k, m+k)), m=1..(n-k)/2)):
%p A212551 seq (seq (T(n, k), k=0..n), n=0..14);
%Y A212551 Columns k=0-10 give: A002865, A083751(n+1), A119907, A212543, A212544, A212545, A212546, A212547, A212548, A212549, A212550.
%Y A212551 Cf. A024786.
%K A212551 nonn,tabl,new
%O A212551 0,11
%A A212551 _Alois P. Heinz_, May 20 2012

%I A212582
%S A212582 8,12,18,20,27,28,30,42,44,45,50,52,63,66,68,70,75,76,78,92,98,99,102,
%T A212582 105,110,114,116,117,124,125,130,138,147,153,154,164,165,170,171,174,
%U A212582 175,182,186,188,190,195,207,230,231,236,238,242,245,246,255,261
%N A212582 Products of exactly three supersingular primes (A002267).
%C A212582 The smallest "triprime" or "3-almost prime" not in this sequence is 148 = 2 * 2 * 37, as 37 is smallest prime which is not a supersingular prime.
%H A212582 T. D. Noe, <a href="/A212582/b212582.txt">Table of n, a(n) for n = 1..680</a> (complete)
%F A212582 i * j * k such that i, j, k are each in A002267, not necessarily distinct.
%t A212582 Union[Times @@@ Tuples[{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}, 3]] (* _T. D. Noe_, May 21 2012 *)
%Y A212582 Cf. A002267, A014612, A108764 (products of exactly two supersingular primes), A212554 (products of supersingular primes).
%K A212582 nonn,easy,fini,new
%O A212582 1,1
%A A212582 _Jonathan Vos Post_, May 21 2012

%I A212556
%S A212556 0,1,2,3,100,101,102,103,110,111,112,113,120,121,122,123,130,131,132,
%T A212556 133,200,201,202,203,210,211,212,213,220,221,222,223,230,231,232,233,
%U A212556 300,301,302,303,310,311,312,313,320,321,322,323,330,331,332,333,10000
%N A212556 Representations of nonnegative integers in base -4: The range of A007608.
%C A212556 Also: Numbers with an odd number of digits, using only digits 0 through 3.
%C A212556 (These numbers (or strings of digits) represent nonnegative integers in base -4, while the same type of numbers with an even number of digits represent negative numbers. See references in A007608 for more information.)
%o A212556 (PARI) {forstep(d=1,3,2, my(u=vector(d,i,10^(d-i))~); forvec(v=vector(d,i,[i==1 & d>1,3]),print1(v*u",")))}
%o A212556 (PARI) is_A212556(n)==(#n=Str(n))%2 & !setminus(Set(Vec(n)),Vec("0123"))
%Y A212556 The same as A007608 sorted in increasing order.
%K A212556 nonn,base,new
%O A212556 1,3
%A A212556 _M. F. Hasler_, May 20 2012

%I A194808
%S A194808 3,0,2,1,3,2,4,4,1,1,3,4,1,1,3,1,3,2,4,2,4,4,1,4,1,1,3,2,4,2,4,4,1,4,
%T A194808 1,1,3,1,3,2,4,4,1,1,3,1,3,1,3,4,1,4,1,2,4,1,3,4,1,4,1,1,3,2,4,2,4,1,
%U A194808 3,4,1,1,3,4,1,1,3,1,3,1,3,4,1,2,4,4,1,1
%N A194808 Twin primes modulo 5.
%F A194808 a(n) = A001097(n) mod 5.
%t A194808 Mod[Select[Prime[Range[300]], PrimeQ[# - 2] || PrimeQ[# + 2] &], 5] (* _T. D. Noe_, May 21 2012 *)
%K A194808 nonn,new
%O A194808 1,1
%A A194808 _Jiri Mazurek_, May 17 2012

%I A212500
%S A212500 1,4,7,36,65,340,615,3220,5825,30500,55175,288900,522625,2736500,
%T A212500 4950375,25920500,46890625,245522500,444154375,2325622500,4207090625,
%U A212500 22028612500,39850134375,208658012500,377465890625,1976437062500,3575408234375
%N A212500 a(n) is the difference between multiples of 5 with even and odd digit sum in base 4 in interval [0,4^n).
%H A212500 V. Shevelev, <a href="http://arxiv.org/abs/0710.3177">On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m</a>
%F A212500 a(1)=1, a(2)=4, a(3)=7, a(4)=36; for n>=5, a(n)=10*a(n-2)-5*a(n-4).
%F A212500 a(n)=0.4*((5+2*sqrt(5))^(n/2)+ (5-2*sqrt(5))^(n/2)) , if n is even, and
%F A212500 a(n)=0.1*((5+2*sqrt(5))^((n-1)/2)*sqrt(30+10*sqrt(5))+(5-2*sqrt(5))^((n-1)/2)*sqrt(30-10*sqrt(5))), if n is odd.
%e A212500 Let n=3. In interval [0,4^3) we have 13 multiples of 5,from which in base 4 only three (namely, 35,50,55)have odd digit sums.Thus a(3)=(13-3)-3=7.
%Y A212500 Cf. A038754, A084990.
%K A212500 nonn,base,easy,new
%O A212500 1,2
%A A212500 _Vladimir Shevelev_ and Peter Moses, May 19 2012

%I A212495
%S A212495 0,2,4,6,8,10,22,24,26,28,30,32,44,46,48,50,52,54,66,68,70,72,74,76,
%T A212495 88,90,92,94,96,98,110,112,114,116,118,120,242,244,246,248,250,252,
%U A212495 264,266,268,270,272,274,286,288,290,292,294,296,308,310,312,314,316
%N A212495 Numbers all of whose base 11 digits are even.
%C A212495 Similar in definition to A033036.
%C A212495 As n increases, it is most likely that A212495(n) < A033036(n), although exceptions to this rule can be found. I conjecture that A212495(n) > A033036(n) for only finitely many values of n.
%e A212495 30 is represented by "28" in base 11. Both digits in this representation are even, thus 30 belongs to the sequence.
%o A212495 (PARI) {is(c) = local(d);while(c != 0, d=c%11; c=(c-d)/11; if(d%2==1, return(0))) ; 1}
%o A212495 for(i=0, 317, if(is(i), print1(i, ", ")))
%Y A212495 Sequence in context: A191749 A038663 A190328 * A198082 A082767 A047932
%K A212495 nonn,base,easy,new
%O A212495 1,2
%A A212495 _Douglas Latimer_, May 18 2012

%I A212557
%S A212557 67,877
%N A212557 Primes p such that (p, p-9) is an irregular pair.
%D A212557 W. Johnson, Irregular primes and Cyclotomic Invariants, Math. Comp., Vol. 29, No. 129, pp. 113-120.
%Y A212557 Special instances of A000928. Variant of A088164 (for the terms of A088164, (p, p-3) is an irregular pair).
%K A212557 nonn,hard,bref,more,new
%O A212557 1,1
%A A212557 _Felix Fröhlich_, May 21 2012

%I A210727
%S A210727 6182296037,6675135377,6798668171,10301484257,12665852879,14922537067,
%T A210727 26348961209,27009595127,30321479693,35572512473,36938181239,
%U A210727 37962662791,45320751701,45999570191,50772316757,52628649973,55745449033,56425976891,57984707603,60553081499
%N A210727 Primes p such that p, p+60, p+120, p+180, p+240 are consecutive primes.
%C A210727 Subsequence of A210683: a(1) = 6182296037 = A210683(146), a(2) = 6675135377 = A210683(166), a(3) = 6798668171 = A210683(175).
%C A210727 Among first 300 terms, the only term p such that p, p+60, p+120, p+180, p+240, p+300 are consecutive primes, is a(172) = 293826343073 (and hence a(173)-a(172)=60 - minimal possible value of first differences).
%H A210727 Zak Seidov, <a href="/A210727/b210727.txt">Table of n, a(n) for n = 1..300</a>
%Y A210727 Cf. A089234, A126771, A210683.
%K A210727 nonn,new
%O A210727 1,1
%A A210727 _Zak Seidov_, May 10 2012

%I A212000
%S A212000 1,3,2,6,5,3,12,11,9,6,20,19,17,14,8,35,34,32,29,23,15,54,53,51,48,42,
%T A212000 34,19,86,85,83,80,74,66,51,32,128,127,125,122,116,108,93,74,42,192,
%U A212000 191,189,186,180,172,157,138,106,64,275,274,272,269,263,255,240
%N A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.
%C A212000 The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.
%C A212000 It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.
%F A212000 T(n,k) = A006128(n) - A006128(k-1).
%F A212000 T(n,k) = Sum_{j=k..n} A138137(j).
%e A212000 For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
%e A212000 --------------------------------------------------------
%e A212000 .  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
%e A212000 --------------------------------------------------------
%e A212000 .  The        Last       Last       Last       The
%e A212000 .  five       four       three      two        last
%e A212000 .  shells     shells     shells     shells     shell
%e A212000 .  of 5       of 5       of 5       of 5       of 5
%e A212000 --------------------------------------------------------
%e A212000 .
%e A212000 .  5          5          5          5          5
%e A212000 .  3+2        3+2        3+2        3+2        3+2
%e A212000 .  4+1        4+1        4+1        4+1          1
%e A212000 .  2+2+1      2+2+1      2+2+1      2+2+1          1
%e A212000 .  3+1+1      3+1+1      3+1+1        1+1          1
%e A212000 .  2+1+1+1    2+1+1+1      1+1+1        1+1          1
%e A212000 .  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
%e A212000 . ---------- ---------- ---------- ---------- ----------
%e A212000 . 20         19         17         14          8
%e A212000 .
%e A212000 So row 5 lists 20, 19, 17, 14, 8.
%e A212000 .
%e A212000 Triangle begins:
%e A212000 1;
%e A212000 3,     2;
%e A212000 6,     5,   3;
%e A212000 12,   11,   9,   6;
%e A212000 20,   19,  17,  14,  8;
%e A212000 35,   34,  32,  29,  23,  15;
%e A212000 54,   53,  51,  48,  42,  34,  19;
%e A212000 86,   85,  83,  80,  74,  66,  51,  32;
%e A212000 128, 127, 125, 122, 116, 108,  93,  74,  42;
%e A212000 192, 191, 189, 186, 180, 172, 157, 138, 106, 64;
%Y A212000 Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.
%Y A212000 Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011
%K A212000 nonn,tabl,new
%O A212000 1,2
%A A212000 _Omar E. Pol_, Apr 26 2012

%I A212010
%S A212010 1,2,3,3,5,6,6,9,11,12,8,14,17,19,20,15,23,29,32,34,35,19,34,42,48,51,
%T A212010 53,54,32,51,66,74,80,83,85,86,42,74,93,108,116,122,125,127,128,64,
%U A212010 106,138,157,172,180,186,189,191,192,83,147,189,221,240
%N A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.
%C A212010 The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.
%C A212010 It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.
%F A212010 T(n,k) = A006128(n) - A006128(n-k).
%F A212010 T(n,k) = Sum_{j=n-k+1..n} A138137(j).
%e A212010 For n = 5 the illustration shows five sets containing the last k shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
%e A212010 --------------------------------------------------------
%e A212010 .  S{5}       S{4-5}     S{3-5}     S{2-5}     S{1-5}
%e A212010 --------------------------------------------------------
%e A212010 .  The        Last       Last       Last       The
%e A212010 .  last       two        three      four       five
%e A212010 .  shell      shells     shells     shells     shells
%e A212010 .  of 5       of 5       of 5       of 5       of 5
%e A212010 --------------------------------------------------------
%e A212010 .
%e A212010 .  5          5          5          5          5
%e A212010 .  3+2        3+2        3+2        3+2        3+2
%e A212010 .    1        4+1        4+1        4+1        4+1
%e A212010 .      1      2+2+1      2+2+1      2+2+1      2+2+1
%e A212010 .      1        1+1      3+1+1      3+1+1      3+1+1
%e A212010 .        1        1+1      1+1+1    2+1+1+1    2+1+1+1
%e A212010 .          1        1+1      1+1+1    1+1+1+1  1+1+1+1+1
%e A212010 . ---------- ---------- ---------- ---------- ----------
%e A212010 .  8         14         17         19         20
%e A212010 .
%e A212010 So row 5 lists 8, 14, 17, 19, 20.
%e A212010 .
%e A212010 Triangle begins:
%e A212010 1;
%e A212010 2,    3;
%e A212010 3,    5,   6;
%e A212010 6,    9,  11,  12;
%e A212010 8,   14,  17,  19,  20;
%e A212010 15,  23,  29,  32,  34,  35;
%e A212010 19,  34,  42,  48,  51,  53,  54;
%e A212010 32,  51,  66,  74,  80,  83,  85,  86;
%e A212010 42,  74,  93, 108, 116, 122, 125, 127, 128;
%e A212010 64, 106, 138, 157, 172, 180, 186, 189, 191, 192;
%Y A212010 Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.
%Y A212010 Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011
%K A212010 nonn,tabl,new
%O A212010 1,2
%A A212010 _Omar E. Pol_, Apr 26 2012

%I A212559
%S A212559 1,1,4,27,244,2745,36966,580111,10399096,209672721,4696872490,
%T A212559 115732052271,3110867569140,90587751885241,2840805169411678,
%U A212559 95450112571474095,3420897993621996016,130266500391456691233,5252293203395848789842,223535386151669737094095,10014286301754519472897900
%N A212559 Number of functions f:{1,2,...,n}->{1,2,...,n} such that every non-recurrent element has at most one preimage.
%C A212559 An element x of {1,2,...,n} is a recurrent element if there exists a positive integer k such that (f^k)(x) = x where f^k is the k-th iteration of functional composition.
%C A212559 The functional digraphs are composed of cycles of rooted trees with every non-root vertex of degree 1 or 2. Cf. A006152.
%F A212559 E.g.f.: 1/(1-A(x)) where A(x) is the e.g.f. for A006152.
%t A212559 nn=20;a=x Exp[x/(1-x)];Range[0,nn]! CoefficientList[Series[1/(1-a),{x,0,nn}],x]
%K A212559 nonn,new
%O A212559 0,3
%A A212559 _Geoffrey Critzer_, May 21 2012

%I A212414
%S A212414 2,12,118,3512,1292274,103071426292,516508833342349371374,
%T A212414 10889035741470030826695916769153787968496,
%U A212414 4168515212543383035248555460312764682619718126289830027967813561362272277233642
%N A212414 Number of conditionally dominated Boolean functions in n variables.
%C A212414 Conditionally dominated Boolean functions arise in the setting of non-cooperatively computable Boolean functions  f: {0,1}^n -> {0,1}; x=(x1,x2,...,xn)-> f(x). Let x' denote x omitting xi and (x',xi) the string that results from inserting xi in x' at the i-th position. A Boolean function f is conditionally dominated if the following two conditions hold:
%C A212414 1. f(x',0) != f(x',1) for some x', that is, the i-th input is relevant.
%C A212414 2. There are binary values xi and y such that f(x',xi)=y for all x'.
%H A212414 Yona Raekow, <a href="/A212414/b212414.txt">Table of n, a(n) for n = 1..12</a>
%H A212414 Y. Raekow and K. Ziegler, A taxonomy of non-cooperatively computable functions, WEWoRC 2011 (<a href="http://www.uni-weimar.de/cms/fileadmin/medien/medsicherheit/WEWoRC2011/files/conference_record3.pdf">conference record</a>)
%F A212414 a(n) = Sum((-2)^(j+1)*C(n, j)*(2^(2^(n-j))-1), j=1..n)-2*n.
%F A212414 a(n) ~ n*2^(2^(n-1)). - _Charles R Greathouse IV_, May 15 2012
%o A212414 (Sage)
%o A212414 result  = 0
%o A212414 for i in range(1,n+1):
%o A212414 ....result += binomial(n,i)*(-1)^(i+1)*2^(i+2^(n-i))
%o A212414 result += (-1)^n-n-1
%o A212414 result *= 2
%o A212414 return result
%o A212414 (PARI) a(n)=sum(j=1,n,(-2)^(j+1)*binomial(n, j)*(1<<2^(n-j)-1))-2*n \\ _Charles R Greathouse IV_, May 15 2012
%K A212414 nonn,new
%O A212414 1,1
%A A212414 _Yona Raekow_, May 15 2012

%I A212418
%S A212418 1,1,1,3,9,54,285,2160,15825,151200,1411095,16329600,185067855,
%T A212418 2514758400,33530101605,523069747200,8020402655265,141228831744000,
%U A212418 2447966414868975,48017802792960000,928344187296100575,20071441567457280000,428190753438433910925
%N A212418 Size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb <--> cba where a<b<c.
%C A212418 Also size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> bac <--> cba where a<b<c.
%H A212418 Alois P. Heinz, <a href="/A212418/b212418.txt">Table of n, a(n) for n = 0..170</a>
%H A212418 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%F A212418 a(n) = 1 for n<3, otherwise: a(2k+1) = (3/2)*k*(k+1)*(2k-1)!, a(2k) = (3/2)*k*(k-1/3)*(2k-2)!-(2k-3)!!.
%e A212418 Contribution from _Alois P. Heinz_, May 21 2012: (Start)
%e A212418 a(3) = 3: {123, 132, 321}.
%e A212418 a(4) = 9: {1234, 1243, 1324, 1342, 1423, 1432, 3214, 4213, 4312}. (End)
%p A212418 a:= proc(n) local k;
%p A212418        k:= iquo(n, 2, 'r');
%p A212418       `if`(n<3, 1, `if`(r=0, (3/2)*k*(k-1/3)*(2*k-2)!
%p A212418        -doublefactorial(2*k-3), (3/2)*k*(k+1)*(2*k-1)!))
%p A212418     end:
%p A212418 seq (a(n), n=0..30);  # _Alois P. Heinz_, May 20 2012
%Y A212418 Cf. A210669.
%K A212418 nonn,new
%O A212418 0,4
%A A212418 _Tom Roby_, May 15 2012

%I A212554
%S A212554 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A212554 27,28,29,30,31,32,33,34,35,36,38,39,40,41,42,44,45,46,47,48,49,50,51,
%U A212554 52,54,55,56,57,58,59,60,62,63,64,65,66,68,69,70,71,72,75,76,77,78,80,81,82,84,85,87,88,90,91,92,93,94,95,96,98,99,100
%N A212554 Products of supersingular primes (A002267).
%C A212554 The initial 1 is included because it has no non-supersingular prime factors.
%C A212554 Many of the early terms divide the order of the monster simple group (see A174670).  The first n such that a(n) does not belong to A174670 is a(204)=289.
%D A212554 J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985.
%D A212554 J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
%D A212554 T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.
%D A212554 A. P. Ogg, Modular functions, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 521-532, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.
%H A212554 T. D. Noe, <a href="/A212554/b212554.txt">Table of n, a(n) for n = 1..10000</a>
%H A212554 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SupersingularPrime.html">Supersingular Prime</a>
%t A212554 ps = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}; fQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, Complement[p, ps] == {}]; Join[{1}, Select[Range[2,1000], fQ]] (* _T. D. Noe_, May 21 2012 *)
%Y A212554 Cf. A002267, A174670, A108764 (products of exactly two supersingular primes).
%K A212554 nonn,new
%O A212554 1,2
%A A212554 _Ben Branman_, May 21 2012

%I A212327
%S A212327 36,62480,141440,1245335,1432640,2286080,6680960,7660880,27931280,
%T A212327 39685376,116636864,179299575,318047135,318523136,358491735,533718135,
%U A212327 709131500,1119849500,1122571695,1814416175,2081125376,3565970135,3991520000,4141021500,4483640576
%N A212327 Numbers n such that A001065(x)*x = n has at least two solutions.
%C A212327 Products of pairs of amicable numbers are members of this sequence.
%e A212327 Example: For n=36, A001065(6)*6 = 36, A001065(9)*9 = 36, therefore 36 is included in this sequence.
%Y A212327 Cf. A212373, A212490, A212489.
%K A212327 nonn,new
%O A212327 1,1
%A A212327 _Naohiro Nomoto_, May 18 2012
%E A212327 a(9)-a(25) from _Donovan Johnson_, May 21 2012

%I A212373
%S A212373 141440,2286080,6680960,1119849500,2081125376,3991520000,4141021500,
%T A212373 9644638208,23664804800,32662630400,37516855536,67994319888,
%U A212373 577461690368,618169892864,627198842816,4132702579824,4949713492400,5025386326400,5579119296000,9013476012156
%N A212373 Numbers n such that A001065(x)*x = n has at least two solutions, where x is an abundant number.
%e A212373 Example: For n=141440, A001065(340)*340 = 141440, A001065(320)*320 = 141440, A001065(340) > 340, A001065(320) > 320, therefore 141440 is included in this sequence.
%Y A212373 Cf. A212327, A005101, A212489, A212490.
%K A212373 nonn,new
%O A212373 1,1
%A A212373 _Naohiro Nomoto_, May 18 2012
%E A212373 a(9)-a(20) from _Donovan Johnson_, May 21 2012

%I A212490
%S A212490 6,336,333312,5418319872,1584858562560
%N A212490 Least number m > 1 such that A000203(x)*x = m has exactly n solutions.
%C A212490 6 = 6.
%C A212490 336 = 6*28*2.
%C A212490 333312 = 6*28*496*2*2.
%C A212490 5418319872 = 6*28*496*8128*2*2*2.
%D A212490 R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.
%e A212490 Example: For n=3, 333312 has exactly 3 solutions, A000203(434)*434 = 333312, A000203(372)*372 = 333312, A000203(336)*336 = 333312, therefore a(3) = 333312.
%Y A212490 Cf. A212489, A212327, A212373.
%K A212490 nonn,more,new
%O A212490 1,1
%A A212490 _Naohiro Nomoto_, May 18 2012
%E A212490 a(5) from _Donovan Johnson_, May 20 2012

%I A212489
%S A212489 1245335,318047135,358491735,533718135,709131500,1122571695,
%T A212489 1814416175,3565970135,4486354631,14336906175,14827262351,22805269551,
%U A212489 36360557831,43971297884,72370166375,99254203895,102204949847,145262865020,156161459559,173741271935,231267964895
%N A212489 Numbers n such that A001065(x)*x = n has at least two solutions, where x is a deficient number.
%e A212489 Example: For n=1245335, A001065(1955)*1955 = 1245335, A001065(2093)*2093 = 1245335, A001065(1955) < 1955, A001065(2093) < 2093, therefore 1245335 is included in this sequence.
%Y A212489 Cf. A005100, A212490, A212373, A212327.
%K A212489 nonn,new
%O A212489 1,1
%A A212489 _Naohiro Nomoto_, May 18 2012
%E A212489 a(6)-a(21) from _Donovan Johnson_, May 21 2012

%I A212530
%S A212530 1,1,1,1,3,5,9,13,19,8,16,1,13,25,4,20,40,17,39,14,36,7,33,2,36,5,39,
%T A212530 2,36,72,39,2,52,11,67,26,84,43,105,62,17,83,38,110,59,2,82,37,127,76,
%U A212530 21,113,54,152,97,40,146,85,22,130,61,175,118,57,181,114
%N A212530 Difference between the sum of the first n primes s(n) and the nearest square <  s(n).
%C A212530 Let  A007504(n) the sum of the first n primes. It is proved that between the numbers A007504(n) and A007504(n+1) there must be a square integer.
%C A212530 The sum of the first n primes is asymptotically equivalent to (1/2)*log(n)*n^2.
%H A212530 Michel Lagneau, <a href="/A212530/b212530.txt">Table of n, a(n) for n = 1..10000</a>
%e A212530 a(5) = 3 because the sum of the 5 primes 2 + 3 + 5 + 7 + 11 = 28, and 28 - 25 = 3.
%p A212530 with(numtheory): for n from 1 to 100 do:s:=sum(‘ithprime(k)’,’k’=1..n):x:=s -floor(sqrt(s-1))^2: printf(`%d, `,x):od:
%Y A212530 Cf. A007504.
%K A212530 nonn,new
%O A212530 1,5
%A A212530 _Michel Lagneau_, May 20 2012

%I A212454
%S A212454 7,13,18,23,29,34,39,44,49,54,60,65,70,75,80,85,90,95,100,105,110,115,
%T A212454 120,125,130,135,140,145,150,156,161,166,171,176,181,186,191,196,201,
%U A212454 206,211,216,221,226,231,236,241,246,251,256,261,266,271,276,281
%N A212454 Ceiling(5n + log(5n)).
%t A212454 Table[Ceiling[5*n + Log[5*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212454 Cf. A000523, A062153, A102572, A212445-A212453.
%K A212454 nonn,easy,new
%O A212454 1,1
%A A212454 _Mohammad K. Azarian_, May 17 2012

%I A212453
%S A212453 6,11,15,19,23,28,32,36,40,44,48,52,56,61,65,69,73,77,81,85,89,93,97,
%T A212453 101,105,109,113,117,121,125,129,133,137,141,145,149,153,158,162,166,
%U A212453 170,174,178,182,186,190,194,198,202,206,210,214,218,222,226,230
%N A212453 Ceiling(4n + log(4n)).
%t A212453 Table[Ceiling[4*n + Log[4*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212453 Cf. A000523, A062153, A102572, A212445-A212454.
%K A212453 nonn,easy,new
%O A212453 1,1
%A A212453 _Mohammad K. Azarian_, May 17 2012

%I A212452
%S A212452 5,8,12,15,18,21,25,28,31,34,37,40,43,46,49,52,55,58,62,65,68,71,74,
%T A212452 77,80,83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,131,
%U A212452 134,137,140,143,146,149,152,156,159,162,165,168,171,174,177,180
%N A212452 Ceiling(3n + log(3n)).
%t A212452 Table[Ceiling[3*n + Log[3*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212452 Cf. A000523, A062153, A102572, A212445-A212454.
%K A212452 nonn,easy,new
%O A212452 1,1
%A A212452 _Mohammad K. Azarian_, May 17 2012

%I A212552
%S A212552 3,13,11,29,15797,53,10949,109912203092239643840221,461,59,
%T A212552 568972471024107865287021434301977158534824481,149,83,173,1693,107,
%U A212552 709,977,269,105649,293,317,2657,179,389,607,1237,137122213,2617,227,509,1049,1097,557,1193,2417,86351
%N A212552 Smallest prime factor of p^p - 1 that is congruent to 1 modulo p where p = prime(n).
%C A212552 Subset of A187023.
%C A212552 If p is a prime, then p^p-1 has at least a prime factor that is congruent to 1 modulo p.
%e A212552 a(4) = 29 because prime(4) = 7 and 7^7 -1 = 823542 = 2 * 3 * 29 * 4733 => 29 == 1 (mod 7).
%p A212552 with(numtheory): for n from 1 to 34 do:i:=0:p:=ithprime(n):x:=p^p -1:y:=factorset(x):n1:=nops(y):for k from 1 to n1 while(i=0) do:z:=y[k]:if irem(z,p)=1 then i:=1: printf ( "%d %d \n",n,z):else fi:od:od:
%t A212552 Table[p=First/@FactorInteger[Prime[n]^Prime[n]-1]; Select[p, Mod[#1, Prime[n]] == 1 &, 1][[1]], {n, 1, 10}]
%Y A212552 Cf. A187023
%K A212552 nonn,new
%O A212552 1,1
%A A212552 _Michel Lagneau_, May 20 2012

%I A212108
%S A212108 0,1,13,62,187,444,885,1616,2677,4227,6326,9163,12742,17468,23167,
%T A212108 30278,38876,49268,61315,75858,92363,111867,134039,159426,187841,
%U A212108 220810,257093,297936,343312,394178,449481,511840,578961,653514,734486
%N A212108 Number of (w,x,y,z) with all terms in {1,...,n} and w*x-y*z<n.
%C A212108 A212108(n)+A211061(n)=n^4.  For a guide to related sequences, see A211795.
%t A212108 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212108 (Do[If[w*x < y*z + n, s = s + 1],
%t A212108 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212108 Map[t[#] &, Range[0, 40]]   (* A212108 *)
%t A212108 (* Peter Moses, Apr 13 2012 *)
%Y A212108 Cf. A211795, A211061.
%K A212108 nonn,new
%O A212108 0,3
%A A212108 _Clark Kimberling_, May 07 2012

%I A212107
%S A212107 0,1,9,48,160,384,798,1468,2517,4041,6172,9063,12870,17746,23907,
%T A212107 31581,40933,52227,65721,81676,100401,122136,147205,175944,208716,
%U A212107 245833,287685,334665,387211,445701,510642,582343,661380,748185,843286
%N A212107 Number of (w,x,y,z) with all terms in {1,...,n} and w >= harmonic mean of {x,y,z}.
%C A212107 A212107(n)+A212106(n)=n^4.  A 4-tuple (w,x,y,z) is counted if 3/w>=1/x+1/y+1/z.  For a guide to related sequences, see A211795.
%t A212107 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212107 (Do[If[w*(y*z + z*x + x*y) >= 3 x*y*z, s = s + 1],
%t A212107 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212107 Map[t[#] &, Range[0, 40]] (* A212107 *)
%t A212107 (* Peter Moses, Apr 13 2012 *)
%Y A212107 Cf. A211795, A212103.
%K A212107 nonn,new
%O A212107 0,3
%A A212107 _Clark Kimberling_, May 04 2012

%I A212106
%S A212106 0,0,7,33,96,241,498,933,1579,2520,3828,5578,7866,10815,14509,19044,
%T A212106 24603,31294,39255,48645,59599,72345,87051,103897,123060,144792,
%U A212106 169291,196776,227445,261580,299358,341178,387196,437736,493050
%N A212106 Number of (w,x,y,z) with all terms in {1,...,n} and w < harmonic mean of {x,y,z}.
%C A212106 A212106(n)+A212107(n)=n^4.  A 4-tuple (w,x,y,z) is counted if 3/w<1/x+1/y+1/z.  For a guide to related sequences, see A211795.
%t A212106 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212106 (Do[If[w*(y*z + z*x + x*y) < 3 x*y*z, s = s + 1],
%t A212106 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212106 Map[t[#] &, Range[0, 40]] (* A212106 *)
%t A212106 (* Peter Moses, Apr 13 2012 *)
%Y A212106 Cf. A211795, A212103.
%K A212106 nonn,new
%O A212106 0,3
%A A212106 _Clark Kimberling_, May 04 2012

%I A212105
%S A212105 0,0,7,45,150,373,768,1437,2479,4002,6120,9010,12786,17661,23821,
%T A212105 31464,40809,52102,65577,81531,100201,121911,146979,175717,208434,
%U A212105 245550,287401,334380,386877,445366,510222,581922,660952,747750,842850
%N A212105 Number of (w,x,y,z) with all terms in {1,...,n} and w > harmonic mean of {x,y,z}.
%C A212105 A212105(n)+A212104(n)=n^4.  A 4-tuple (w,x,y,z) is counted if 3/w>1/x+1/y+1/z.  For a guide to related sequences, see A211795.
%t A212105 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212105 (Do[If[w*(y*z + z*x + x*y) > 3 x*y*z, s = s + 1],
%t A212105 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212105 Map[t[#] &, Range[0, 40]] (* A212105 *)
%t A212105 (* Peter Moses, Apr 13 2012 *)
%Y A212105 Cf. A211795, A212103.
%K A212105 nonn,new
%O A212105 0,3
%A A212105 _Clark Kimberling_, May 04 2012

%I A212104
%S A212104 0,1,9,36,106,252,528,964,1617,2559,3880,5631,7950,10900,14595,19161,
%T A212104 24727,31419,39399,48790,59799,72570,87277,104124,123342,145075,
%U A212104 169575,197061,227779,261915,299778,341599,387624,438171,493486
%N A212104 Number of (w,x,y,z) with all terms in {1,...,n} and w <= harmonic mean of {x,y,z}.
%C A212104 A212104(n)+A212105(n)=n^4.  A 4-tuple (w,x,y,z) is counted if 3/w<=1/x+1/y+1/z.  For a guide to related sequences, see A211795.
%t A212104 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212104 (Do[If[w*(y*z + z*x + x*y) <= 3 x*y*z, s = s + 1],
%t A212104 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212104 Map[t[#] &, Range[0, 50]] (* A212104 *)
%t A212104 FindLinearRecurrence[%]
%t A212104 (* Peter Moses, Apr 13 2012 *)
%Y A212104 Cf. A211795, A212103.
%K A212104 nonn,new
%O A212104 0,3
%A A212104 _Clark Kimberling_, May 04 2012

%I A212451
%S A212451 3,6,8,11,13,15,17,19,21,23,26,28,30,32,34,36,38,40,42,44,46,48,50,52,
%T A212451 54,56,58,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,95,97,99,
%U A212451 101,103,105,107,109,111,113,115,117,119,121,123,125,127,129
%N A212451 Ceiling(2n + log(2n)).
%t A212451 Table[Ceiling[2*n + Log[2*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212451 Cf. A000523, A062153, A102572, A212445-A212454.
%K A212451 nonn,easy,new
%O A212451 1,1
%A A212451 _Mohammad K. Azarian_, May 17 2012

%I A212445
%S A212445 1,2,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,24,25,26,27,28,
%T A212445 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,
%U A212445 52,53,54,55,56,57,59,60,61,62,63,64,65,66,67,68,69,70
%N A212445 Floor(n + log(n)).
%t A212445 Table[Floor[n + Log[n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212445 Cf. A000523, A062153, A102572, A212446, A212447, A212448, A212449, A212450, A212451, A212452, A212453, A212454.
%K A212445 nonn,easy,new
%O A212445 1,2
%A A212445 _Mohammad K. Azarian_, May 17 2012

%I A212450
%S A212450 1,3,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,29,
%T A212450 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
%U A212450 53,54,55,56,57,58,60,61,62,63,64,65,66,67,68,69,70,71
%N A212450 Ceiling(n + log(n)).
%t A212450 Table[Ceiling[n + Log[n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212450 Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212449, A212451, A212452, A212453, A212454.
%K A212450 nonn,easy,new
%O A212450 1,2
%A A212450 _Mohammad K. Azarian_, May 17 2012

%I A212449
%S A212449 6,12,17,22,28,33,38,43,48,53,59,64,69,74,79,84,89,94,99,104,109,114,
%T A212449 119,124,129,134,139,144,149,155,160,165,170,175,180,185,190,195,200,
%U A212449 205,210,215,220,225,230,235,240,245,250,255,260,265,270,275,280,285
%N A212449 Floor(5n + log(5n)).
%t A212449 Table[Floor[5*n + Log[5*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212449 Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212448, A212450, A212451, A212452, A212453, A212454.
%K A212449 nonn,easy,new
%O A212449 1,1
%A A212449 _Mohammad K. Azarian_, May 17 2012

%I A212448
%S A212448 5,10,14,18,22,27,31,35,39,43,47,51,55,60,64,68,72,76,80,84,88,92,96,
%T A212448 100,104,108,112,116,120,124,128,132,136,140,144,148,152,157,161,165,
%U A212448 169,173,177,181,185,189,193,197,201,205,209,213,217,221,225,229
%N A212448 Floor(4n + log(4n)).
%t A212448 Table[Floor[4*n + Log[4*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212448 Cf. A000523, A062153, A102572, A212445, A212446, A212447, A212449, A212450, A212451, A212452, A212453, A212454.
%K A212448 nonn,easy,new
%O A212448 1,1
%A A212448 _Mohammad K. Azarian_, May 17 2012

%I A212447
%S A212447 4,7,11,14,17,20,24,27,30,33,36,39,42,45,48,51,54,57,61,64,67,70,73,
%T A212447 76,79,82,85,88,91,94,97,100,103,106,109,112,115,118,121,124,127,130,
%U A212447 133,136,139,142,145,148,151,155,158,161,164,167,170,173,176,179
%N A212447 Floor(3n + log(3n)).
%t A212447 Table[Floor[3*n + Log[3*n]], {n, 100}] (* _T. D. Noe_, May 21 2012 *)
%Y A212447 Cf. A000523, A062153, A102572, A212445, A212446, A212448, A212449, A212450, A212451, A212452, A212453, A212454.
%K A212447 nonn,easy,new
%O A212447 1,1
%A A212447 _Mohammad K. Azarian_, May 17 2012

%I A212419
%S A212419 1,1,1,4,21,116,713,5030,40301,362852,3628744,39916716,479001426,
%T A212419 6227020536,87178290639,1307674367142,20922789886141,355687428093140,
%U A212419 6402373705721708,121645100408822276,2432902008176618342,51090942171709406408,1124000727777607604418
%N A212419 Size of the equivalence class of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb <--> bac <--> cba, where a<b<c.
%C A212419 Pierrot, Rossin, and West were first to give a formula and the alternate characterization: all permutations in S_n except the alternating permutations in which the elements in odd positions form a decreasing sequence, and the elements in even positions also form a decreasing sequence.
%H A212419 Alois P. Heinz, <a href="/A212419/b212419.txt">Table of n, a(n) for n = 0..170</a>
%H A212419 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%H A212419 A. Pierrot, D. Rossin, and J. West, <a href="http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAO0167/3638">Adjacent transformations in permutations</a>, FPSAC 2011, Discrete Math. Theor. Comput. Sci. Proc., 2011.
%F A212419 a(n) = 1 for n<3, otherwise: a(n) = n!-C([(n-1)/2]-C([n/2]), where [x] is the floor function and C(n) denotes the n-th Catalan number (A000108).
%p A212419 C:= n-> binomial(2*n, n)/(n+1):
%p A212419 a:= n-> `if`(n<3, 1, n!-C(floor((n-1)/2))-C(floor(n/2))):
%p A212419 seq (a(n), n=0..30);  # _Alois P. Heinz_, May 20 2012
%K A212419 nonn,new
%O A212419 0,4
%A A212419 _Tom Roby_, May 15 2012

%I A212446
%S A212446 2,5,7,10,12,14,16,18,20,22,25,27,29,31,33,35,37,39,41,43,45,47,49,51,
%T A212446 53,55,57,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,
%U A212446 100,102,104,106,108,110,112,114,116,118,120,122,124,126,128
%N A212446 Floor(2n + log(2n)).
%t A212446 Table[Floor[2*n + Log[2*n]], {n, 100}] (* T. D. Noe, May 21 2012 *)
%Y A212446 Cf. A000523, A062153, A102572, A212445, A212447, A212448, A212449, A212450, A212451, A212452, A212453, A212454.
%K A212446 nonn,easy,new
%O A212446 1,1
%A A212446 _Mohammad K. Azarian_, May 17 2012

%I A182523
%S A182523 2,6,170,9520,874902,118950678,22370367448,5550123527520
%V A182523 -2,-6,-170,-9520,-874902,-118950678,-22370367448,-5550123527520
%N A182523 Rademacher's sequence C_{011}(N) times (2n)!, where C_{011}(N) is the coefficient of 1/(q-1) in the partial fraction decomposition of 1/((1-q)(1-q^2)...(1-q^N)).
%C A182523 Hans Rademacher conjectured that C_{011}(N) converge to -0.292927573960. This conjecture is false.
%H A182523 A. Sills and D. Zeilberger, <a href="http://arxiv.org/abs/1110.4932">Rademacher's Infinite Partial Fraction Conjecture is False</a>, arXiv:1110.4932v1 [math.NT]
%H A182523 A. Sills and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/hans.html">Rademacher's Infinite Partial Fraction Conjecture is (almost certainly) False</a>, Oct 21 2011.
%H A182523 A. Sills and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/HANS">HANS (maple package)</a>
%F A182523 See above article for an efficient recurrence.
%e A182523 For n=1, the coefficient of 1/(q-1) in the partial fraction decomposition of 1/(1-q) is -1, multiplied by 2! this gives -2.
%p A182523 See above link to HANS (maple package).
%K A182523 sign,more,new
%O A182523 1,1
%A A182523 _Shalosh B. Ekhad_, May 03 2012

%I A212103
%S A212103 0,1,2,3,10,11,30,31,38,39,52,53,84,85,86,117,124,125,144,145,200,225,
%T A212103 226,227,282,283,284,285,334,335,420,421,428,435,436,491,546,547,548,
%U A212103 555,634,635,726,727,758,837,838,839,936,937,956,957,970,971
%N A212103 Number of (w,x,y,z) with all terms in {1,...,n} and w = harmonic mean of {x,y,z}.
%C A212103 Also, the number of (w,x,y,z) with all terms in {1,...,n} and H(w,x,y)=H(w,x,y,z) where H denotes harmonic mean.  For a guide to related sequences, see A211795.
%e A212103 a(4) counts these:  (1,1,1,1), (2,1,4,4), (2,2,2,2), (2,4,1,4), (2,4,4,1), (3,2,4,4), (3,3,3,3), (3,4,2,4), (3,4,4,2), (4,4,4,4); e.g., (3,2,4,4) is included because it satisfies 3/w=1/x+1/y+1/z.
%t A212103 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212103 (Do[If[w*(y*z + z*x + x*y) == 3 x*y*z, s = s + 1],
%t A212103 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212103 Map[t[#] &, Range[0, 60]] (* A212103 *)
%t A212103 (* Peter Moses, Apr 13 2012 *)
%Y A212103 Cf. A211795.
%K A212103 nonn,new
%O A212103 0,3
%A A212103 _Clark Kimberling_, May 03 2012

%I A212102
%S A212102 0,0,0,1,4,4,11,11,14,15,18,18,34,34,34,56,59,59,78,78,105,112,112,
%T A212102 112,143,143,143,144,177,177,235,235,238,245,245,269,318,318,318,319,
%U A212102 367,367,416,416,422,471,471,471,517,517,520,521,530,530,549,561
%N A212102 Number of (w,x,y,z) with all terms in {1,...,n} and 1/w=1/x+1/y+1/z.
%C A212102 For a guide to related sequences, see A211795.
%e A212102 a(4) counts these: (1,3,3,3), (1,2,4,4), (1,4,2,4), (1,4,4,2).
%t A212102 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212102 (Do[If[w*(y*z + z*x + x*y) == x*y*z, s = s + 1],
%t A212102 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212102 Map[t[#] &, Range[0, 60]] (* A212102 *)
%t A212102 (* Peter Moses, Apr 13 2012 *)
%Y A212102 Cf. A211795.
%K A212102 nonn,new
%O A212102 0,5
%A A212102 _Clark Kimberling_, May 03 2012

%I A212101
%S A212101 0,1,4,9,20,29,42,55,80,109,132,153,196,221,254,287,356,389,460,497,
%T A212101 568,617,670,715,808,897,960,1057,1156,1213,1306,1367,1512,1589,1672,
%U A212101 1749,1964,2037,2130,2223,2376,2457,2580,2665,2824,2997,3110
%N A212101 Number of (w,x,y,z) with all terms in {1,...,n} and w*x^2=y*z^2.
%C A212101 For a guide to related sequences, see A211795.
%e A212101 The four (w,x,y,z) counted by a(2): (1,1,1,1), (1,2,1,2), (2,1,2,1), (2,2,2,2).
%t A212101 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212101 (Do[If[w*x^2 == y*z^2, s = s + 1],
%t A212101 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212101 Map[t[#] &, Range[0, 60]] (* A212101 *)
%t A212101 (* Peter Moses, Apr 13 2012 *)
%Y A212101 Cf. A211795.
%K A212101 nonn,new
%O A212101 0,3
%A A212101 _Clark Kimberling_, May 03 2012

%I A212099
%S A212099 0,0,1,9,35,89,192,367,645,1047,1620,2395,3414,4749,6435,8518,11079,
%T A212099 14171,17876,22272,27409,33396,40290,48249,57265,67548,79146,92127,
%U A212099 106708,122880,140876,160757,182694,206791,233160,262032,293445
%N A212099 Number of (w,x,y,z) with all terms in {1,...,n} and w^3>x^3+y^3+z^3.
%C A212099 A212099(n)+A212098(n)=n^4.  For a guide to related sequences, see A211795.
%t A212099 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212099 (Do[If[w^3 > x^3 + y^3 + z^3, s = s + 1],
%t A212099 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212099 Map[t[#] &, Range[0, 40]] (* A212099 *)
%t A212099 (* Peter Moses, Apr 13 2012 *)
%Y A212099 Cf. A211795.
%K A212099 nonn,new
%O A212099 0,4
%A A212099 _Clark Kimberling_, May 03 2012

%I A212098
%S A212098 0,1,15,72,221,536,1104,2034,3451,5514,8380,12246,17322,23812,31981,
%T A212098 42107,54457,69350,87100,108049,132591,161085,193966,231592,274511,
%U A212098 323077,377830,439314,507948,584401,669124,762764,865882,979130
%N A212098 Number of (w,x,y,z) with all terms in {1,...,n} and w^3<=x^3+y^3+z^3.
%C A212098 A212098(n)+A212099(n)=n^4.  For a guide to related sequences, see A211795.
%t A212098 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212098 (Do[If[w^3 <= x^3 + y^3 + z^3, s = s + 1],
%t A212098 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212098 Map[t[#] &, Range[0, 40]] (* A212098 *)
%t A212098 (* Peter Moses, Apr 13 2012 *)
%o A212098 (PARI) A212098(n)={my(s=0,c=[6,3,1]);forvec(v=vector(4,i,if(i>1,[1,n],[-n,-1])),sum(i=1,4,v[i]^3)>=0&s+=c[1+(v[2]==v[3])+(v[3]==v[4])],1);s} /* not very efficient */ _M. F. Hasler_, May 20 2012
%Y A212098 Cf. A211795.
%K A212098 nonn,new
%O A212098 0,3
%A A212098 _Clark Kimberling_, May 03 2012

%I A212494
%S A212494 0,1,2,3,10300,10301,10302,10303,10200,10201,10202,10203,10100,10101,
%T A212494 10102,10103,10000,10001,10002,10003,20300,20301,20302,20303,20200,
%U A212494 20201,20202,20203,20100,20101,20102,20103
%N A212494 Base 2i representation of nonnegative integers.
%C A212494 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 quater-imaginary, dispenses with the need to represent the real and imaginary parts of a complex number separately. (The term "quater-imaginary" appears in Knuth's landmark book on computer programming).
%C A212494 Quater-imaginary, based on the powers of 2i (twice the imaginary unit), uses the digits 0, 1, 2, 3. For purely real positive integers, the quater-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.
%C A212494 To obtain the base 2i representation of a complex number a+bi, do as above for the real part, then again for the real part of 2i*(a+bi) = -2b+2ai, giving the digits corresponding to the odd-indexed powers of 2i.
%C A212494 Omitting digits for odd powers of 2i (all 0's for the imaginary parts) (e.g. 20300 --> 230) gives A007608 (nonnegative integers in base -4).
%D A212494 Donald Knuth, The Art of Computer Programming. Volume 2, 2nd Edition. Reading, Massachussetts: Addison-Wesley (1981): 189
%H A212494 Joerg Arndt, <a href="/A212494/b212494.txt">Table of n, a(n) for n = 0..1000</a>
%H A212494 Joerg Arndt, <a href="http://www.jjj.de/fxt/demo/bits/#radix-2i">Radix 2i</a>
%H A212494 Donald Knuth, <a href="http://dl.acm.org/citation.cfm?id=367233">An imaginary number system</a>, Communications of the ACM 3 (4), April 1960, pp. 245-247.
%H A212494 OEIS Wiki, <a href="/wiki/Quater-imaginary_base">Quater-imaginary base</a>
%H A212494 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quater-imaginary_base">Quater-imaginary base</a>
%e A212494 a(5) = 10301 because 5 = 1*(2i)^4+3*(2i)^2+1*(2i)^0 = 1*16+3*(-4)+1*1
%Y A212494 Cf. A212542 (Base 2i representation of negative integers).
%Y A212494 Cf. A177505.
%Y A212494 Cf. A007608 (Nonnegative integers in base -4).
%K A212494 nonn,base,new
%O A212494 0,3
%A A212494 _Daniel Forgues_, May 18 2012

%I A182320
%S A182320 2,5,7,11,13,17,37,41,67,97,101,103,107,191,193,223,227,277,307,311,
%T A182320 347,457,461,613,641,773,821,823,853,857,877,881,1013,1087,1091,1277,
%U A182320 1297,1301,1373,1423,1427,1447,1481,1483,1487,1607,1663,1693,1811,1867,1871
%N A182320 Primes p = prime(n) such that the equation prime(n+k) - prime(n) = 6^(k-1) has at least one solution, k>0.
%C A182320 The first term having k=5 as solution is larger than 10^9. - _M. F. Hasler_, May 20 2012
%e A182320 a(1) = 2 = prime(1) = prime(1+1) - 6^(1-1) = 3 - 1 is the only term with k=1 as solution.
%e A182320 a(2) = 5 = prime(3) = prime(3+2) - 6^(2-1) = 11 - 6.
%e A182320 a(26) = 773 =  prime(137) =  prime(137+3) - 6^2 = 809 - 36 is the first term having k=3 as smallest solution.
%e A182320 10915517 = prime(721294) = prime(721294+4) - 6^3 = 10915733 - 216 is the first term having k=4 as solution. - _M. F. Hasler_, May 20 2012
%o A182320 (PARI) is_A182320(p)={isprime(p)||return;my(q=p);for(k=0,9,p+6^k==(q=nextprime(q+1))&return(1))}  \\ _M. F. Hasler_, May 20 2012
%Y A182320 Cf. A001223, A182135, A182217.
%K A182320 nonn,new
%O A182320 1,1
%A A182320 _Gerasimov Sergey_, Apr 24 2012

%I A182217
%S A182217 2,3,43,73,151,157,163,181,277,337,367,373,433,487,601,631,643,727,
%T A182217 757,811,823,937,967,1093,1213,1471,1483,1543,1567,1693,1873,2083,
%U A182217 2137,2281,2341,2383,2647,2671,2953,3307,3313,3517,3607,3847,4003,4441,4447
%N A182217 Primes p = prime(n) such that there is k>0 for which prime(n+k) = prime(n) + 4^(k-1).
%e A182217 2=prime(1+1)-4^(1-1)=3-1, 3=prime(2+2)-4^(2-1)=7-4, 43=prime(14+3)-4^(3-1)=59-16, 73=prime(21+3)-4^(3-1)=89-16.
%o A182217 (PARI) is_A182217(p)={isprime(p) || return; my(q=p); for(k=0,9, p+4^k==(q=nextprime(q+1)) & return(1))}  \\ _M. F. Hasler_, May 20 2012
%o A182217 (PARI) for(n=1,9999,for(k=1,9,prime(n+k)-prime(n)==4^(k-1)&!print1(prime(n)",")&break))  \\ _M. F. Hasler_, May 20 2012
%Y A182217 Cf. A001223, A182135.
%K A182217 nonn,new
%O A182217 1,1
%A A182217 _Gerasimov Sergey_, Apr 19 2012

%I A212499
%S A212499 1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100,101,102,103,104,105,
%T A212499 106,107,108,109,110,120,130,140,150,160,170,180,190,200,201,202,203,
%U A212499 204,205,206,207,208,209,210,220,230,240,250,260,270,280,290,300,301
%N A212499 Numbers n such that n divides the product of digits of n.
%C A212499 Every integer except zero divides zero.
%C A212499 a(n+9) = A011540(n+1).
%C A212499 A050720(2*n) = number of terms of length n for n >= 2.
%H A212499 Arkadiusz Wesolowski, <a href="/A212499/b212499.txt">Table of n, a(n) for n = 1..1000</a>
%t A212499 Union[Range[9], Select[Range[10, 301], DigitCount[#, 10, 0] > 0 &]]
%t A212499 Select[Range[301], Divisible[Product[i, {i, IntegerDigits[#]}], #] &]
%Y A212499 Cf. A049101.
%K A212499 base,easy,nonn,new
%O A212499 1,2
%A A212499 _Arkadiusz Wesolowski_, May 19 2012

%I A212096
%S A212096 0,0,0,0,0,0,6,6,6,12,12,12,18,18,18,18,18,18,30,36,42,42,42,42,48,54,
%T A212096 54,60,66,72,78,78,78,78,78,78,90,90,96,96,102,114,120,120,126,132,
%U A212096 144,144,150,150,156,156,156,162,180,180,186,192,204,204,216,216
%N A212096 Number of (w,x,y,z) with all terms in {1,...,n} and w^3=x^3+y^3+z^3.
%C A212096 Every term is divisible by 6.  For a guide to related sequences, see A211795.
%t A212096 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212096 (Do[If[w^3 == x^3 + y^3 + z^3, s = s + 1],
%t A212096 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212096 Map[t[#] &, Range[0, 70]] (* A212096 *)
%t A212096 %/6  (* integers *)
%t A212096 (* Peter Moses, Apr 13 2012 *)
%Y A212096 Cf. A211795.
%K A212096 nonn,new
%O A212096 0,7
%A A212096 _Clark Kimberling_, May 02 2012

%I A212093
%S A212093 0,1,15,76,234,565,1161,2145,3644,5825,8854,12940,18300,25173,33814,
%T A212093 44524,57587,73346,92127,114312,140272,170421,205198,245043,290413,
%U A212093 341832,399774,464801,537425,618289,707923,807009,916110,1035937
%N A212093 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<=x^2+y^2+z^2.
%C A212093 A212093(n)+A212094(n)=n^4.  For a guide to related sequences, see A211795.
%t A212093 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212093 (Do[If[w^2 <= x^2 + y^2 + z^2, s = s + 1],
%t A212093 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212093 Map[t[#] &, Range[0, 50]] (* A212093 *)
%t A212093 (* Peter Moses, Apr 13 2012 *)
%Y A212093 Cf. A211795.
%K A212093 nonn,new
%O A212093 0,3
%A A212093 _Clark Kimberling_, May 02 2012

%I A212094
%S A212094 0,0,1,5,22,60,135,256,452,736,1146,1701,2436,3388,4602,6101,7949,
%T A212094 10175,12849,16009,19728,24060,29058,34798,41363,48793,57202,66640,
%U A212094 77231,88992,102077,116512,132466,149984,169206,190205,213109,238015
%N A212094 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>x^2+y^2+z^2.
%C A212094 A212094(n)+A212093(n)=n^4.  For a guide to related sequences, see A211795.
%t A212094 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212094 (Do[If[w^2 > x^2 + y^2 + z^2, s = s + 1],
%t A212094 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212094 Map[t[#] &, Range[0, 50]] (* A212094 *)
%t A212094 (* Peter Moses, Apr 13 2012 *)
%Y A212094 Cf. A211795.
%K A212094 nonn,new
%O A212094 0,4
%A A212094 _Clark Kimberling_, May 02 2012

%I A212097
%S A212097 0,1,15,72,221,536,1098,2028,3445,5502,8368,12234,17304,23794,31963,
%T A212097 42089,54439,69332,87070,108013,132549,161043,193924,231550,274463,
%U A212097 323023,377776,439254,507882,584329,669046,762686,865804,979052
%N A212097 Number of (w,x,y,z) with all terms in {1,...,n} and w^3<x^3+y^3+z^3.
%C A212097 A212097(n)+A212100(n)=4^n.  For a guide to related sequences, see A211795.
%t A212097 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212097 (Do[If[w^3 < x^3 + y^3 + z^3, s = s + 1],
%t A212097 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212097 Map[t[#] &, Range[0, 40]] (* A212097 *)
%t A212097 (* Peter Moses, Apr 13 2012 *)
%Y A212097 Cf. A211795.
%K A212097 nonn,new
%O A212097 0,3
%A A212097 _Clark Kimberling_, May 03 2012

%I A212100
%S A212100 0,0,1,9,35,89,198,373,651,1059,1632,2407,3432,4767,6453,8536,11097,
%T A212100 14189,17906,22308,27451,33438,40332,48291,57313,67602,79200,92187,
%U A212100 106774,122952,140954,160835,182772,206869,233238,262110,293535
%N A212100 Number of (w,x,y,z) with all terms in {1,...,n} and w^3>=x^3+y^3+z^3.
%C A212100 A212100(n)+A212097(n)=n^4.  For a guide to related sequences, see A211795.
%t A212100 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212100 (Do[If[w^3 >= x^3 + y^3 + z^3, s = s + 1],
%t A212100 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212100 Map[t[#] &, Range[0, 40]] (* A212100 *)
%t A212100 (* Peter Moses, Apr 13 2012 *)
%Y A212100 Cf. A211795.
%K A212100 nonn,new
%O A212100 0,4
%A A212100 _Clark Kimberling_, May 03 2012

%I A212542
%S A212542 103,102,101,100,203,202,201,200,303,302,301,300,1030003,1030002,
%T A212542 1030001,1030000,1030103,1030102,1030101,1030100,1030203,1030202,
%U A212542 1030201,1030200,1030303,1030302,1030301,1030300,1020003,1020002,1020001,1020000,1020103,1020102,1020101,1020100,1020203,1020202,1020201,1020200,1020303
%N A212542 Base 2i representation of negative integers.
%C A212542 Omitting digits for odd powers of 2i (all 0's for the imaginary parts) (e.g. 1030003 --> 1303) gives A212526 (negative integers in base -4).
%H A212542 Joerg Arndt, <a href="/A212542/b212542.txt">Table of n, a(n) for n = 1..1000</a>
%e A212542 a(13) = 1030003: 1*(2*i)^6 + 0 + 3*(2*i)^4 + 0 + 0 + 0 + 3*(2*i)^0 = -64 + 48 + 3 = -13.
%p A212542 a:= proc(n) local d, i, l, m;
%p A212542       m:= n; l:= NULL;
%p A212542       for i from 0 while m>0 do
%p A212542         d:= irem(m, 4, 'm');
%p A212542         if irem (i, 2)=0 and d>0 then d:= 4-d; m:= m+1 fi;
%p A212542         l:= d, 0, l
%p A212542       od; parse(cat(l))/10
%p A212542     end:
%p A212542 seq (a(n), n=1..60); # _Alois P. Heinz_, May 20 2012
%Y A212542 Cf. A212494 (base 2i representation of nonnegative integers).
%K A212542 base,nonn,new
%O A212542 1,1
%A A212542 _Joerg Arndt_, May 20 2012
%E A212542 Minor edit

%I A212526
%S A212526 13,12,11,10,23,22,21,20,33,32,31,30,1303,1302,1301,1300,1313,1312,
%T A212526 1311,1310,1323,1322,1321,1320,1333,1332,1331,1330,1203,1202,1201,
%U A212526 1200,1213,1212,1211,1210,1223,1222,1221,1220
%N A212526 Negative integers in base -4.
%C A212526 Interleaving with zeros gives A212542 (base 2i representation of negative integers).
%C A212526 More precisely, a(n) is the representation of -n in base -4. - _M. F. Hasler_, May 21 2012
%H A212526 Joerg Arndt, <a href="/A212526/b212526.txt">Table of n, a(n) for n = 1..1000</a>
%e A212526 a(13)=1303:  1*(-4)^3 + 3*(-4)^2 + 0*(-4)^1 + 3*(-4)^0 = -64 + 48 +3 = -13.
%p A212526 a:= proc(n) local d, i, l, m;
%p A212526       m:= n;
%p A212526       l:= NULL;
%p A212526       for i from 0 while m>0 do
%p A212526         d:= irem(m, 4, 'm');
%p A212526         if irem (i, 2)=0 and d>0 then d:= 4-d; m:= m+1 fi;
%p A212526         l:= d, l
%p A212526       od; parse(cat(l))
%p A212526     end:
%p A212526 seq (a(n), n=1..60);  # _Alois P. Heinz_, May 20 2012
%o A212526 (PARI) A212526(n,s="")={n=-n;until(!n\=-4,s=Str(n%-4,s));eval(s)}  \\ _M. F. Hasler_, May 21 2012
%Y A212526 Cf. A007608 (Nonnegative integers in base -4).
%K A212526 nonn,base,new
%O A212526 1,1
%A A212526 _Joerg Arndt_, May 20 2012

%I A212529
%S A212529 11,10,1101,1100,1111,1110,1001,1000,1011,1010,110101,110100,110111,
%T A212529 110110,110001,110000,110011,110010,111101,111100,111111,111110,
%U A212529 111001,111000,111011,111010,100101,100100,100111,100110,100001,100000,100011,100010,101101,101100,101111,101110,101001,101000,101011,101010,11010101
%N A212529 Negative numbers in base -2.
%H A212529 Joerg Arndt, <a href="/A212529/b212529.txt">Table of n, a(n) for n = 1..1000</a>
%p A212529 a:= proc(n) local d, i, l, m;
%p A212529       m:= n; l:= NULL;
%p A212529       for i from 0 while m>0 do
%p A212529         d:= irem(m, 2, 'm');
%p A212529         if d=1 and irem (i, 2)=0 then m:= m+1 fi;
%p A212529         l:= d, l
%p A212529       od; parse(cat(l))
%p A212529     end:
%p A212529 seq (a(n), n=1..100);  # _Alois P. Heinz_, May 20 2012
%Y A212529 Cf. A039724 (nonnegative numbers in base -2).
%Y A212529 Cf. A007608 (nonnegative numbers in base -4), A212526 (negative numbers in base -4).
%K A212529 nonn,base,new
%O A212529 1,1
%A A212529 _Joerg Arndt_, May 20 2012

%I A212095
%S A212095 0,0,1,8,25,63,141,268,464,760,1170,1734,2472,3430,4650,6164,8012,
%T A212095 10247,12933,16108,19827,24192,29199,34957,41525,48967,57382,66859,
%U A212095 77456,89235,102335,116794,132748,150314,169545,190574,213490,238420
%N A212095 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x^2+y^2+z^2.
%C A212095 A2120945(n)+A212092(n)=n^4.  For a guide to related sequences, see A211795.
%t A212095 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212095 (Do[If[w^2 >= x^2 + y^2 + z^2, s = s + 1],
%t A212095 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212095 Map[t[#] &, Range[0, 50]] (* A212095 *)
%t A212095 (* Peter Moses, Apr 13 2012 *)
%Y A212095 Cf. A211795.
%K A212095 nonn,new
%O A212095 0,4
%A A212095 _Clark Kimberling_, May 02 2012

%I A212092
%S A212092 0,1,15,73,231,562,1155,2133,3632,5801,8830,12907,18264,25131,33766,
%T A212092 44461,57524,73274,92043,114213,140173,170289,205057,244884,290251,
%U A212092 341658,399594,464582,537200,618046,707665,806727,915828,1035607
%N A212092 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<x^2+y^2+z^2.
%C A212092 A212092(n)+A212095(n)=n^4.
%C A212092 For a guide to related sequences, see A211795.
%t A212092 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212092 (Do[If[w^2 < x^2 + y^2 + z^2, s = s + 1],
%t A212092 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212092 Map[t[#] &, Range[0, 50]] (* A212092 *)
%t A212092 (* Peter Moses, Apr 13 2012 *)
%Y A212092 Cf. A211795.
%K A212092 nonn,new
%O A212092 0,3
%A A212092 _Clark Kimberling_, May 02 2012

%I A212540
%S A212540 1,5,12,24,33,69,83,116,145,188,193,290,293,332,428,496,497,606,607,
%T A212540 772,840,857,858,1079,1098,1109,1153,1277,1278,1626,1627,1723,1750,
%U A212540 1757,1832,2100,2101,2106,2130,2541,2542,2784,2785,2843,3111,3115,3116,3462,3477
%N A212540 Number of nondecreasing sequences of 7 1..n integers with every element dividing the sequence sum
%C A212540 Row 7 of A212536
%H A212540 R. H. Hardin, <a href="/A212540/b212540.txt">Table of n, a(n) for n = 1..210</a>
%e A212540 Some solutions for n=8
%e A212540 ..2....2....1....1....1....2....1....1....2....1....1....1....2....1....3....1
%e A212540 ..2....2....2....1....1....2....1....6....2....2....1....2....3....3....3....2
%e A212540 ..4....2....2....5....2....2....1....7....5....5....3....3....3....4....3....2
%e A212540 ..4....2....2....5....2....3....1....7....5....8....3....4....3....4....5....3
%e A212540 ..4....4....7....6....4....3....2....7....5....8....4....4....3....4....5....4
%e A212540 ..8....6....7....6....5....4....2....7....5....8....4....4....4....4....5....6
%e A212540 ..8....6....7....6....5....8....4....7....6....8....8....6....6....4....6....6
%K A212540 nonn,new
%O A212540 1,2
%A A212540 _R. H. Hardin_ May 20 2012

%I A212539
%S A212539 1,4,8,15,21,40,44,64,76,103,104,148,149,165,197,229,230,271,272,343,
%T A212539 362,367,368,449,457,462,478,521,522,639,640,677,685,688,707,800,801,
%U A212539 804,812,937,938,1011,1012,1026,1094,1097,1098,1204,1208,1241,1246,1262,1263
%N A212539 Number of nondecreasing sequences of 6 1..n integers with every element dividing the sequence sum
%C A212539 Row 6 of A212536
%H A212539 R. H. Hardin, <a href="/A212539/b212539.txt">Table of n, a(n) for n = 1..210</a>
%e A212539 Some solutions for n=8
%e A212539 ..1....2....1....1....2....1....1....5....1....2....1....1....1....6....2....2
%e A212539 ..1....2....1....1....2....1....1....5....2....4....1....1....1....6....3....2
%e A212539 ..2....2....1....2....4....1....2....5....4....4....1....4....1....6....3....3
%e A212539 ..2....6....1....2....4....1....2....5....4....4....2....6....1....6....4....3
%e A212539 ..3....6....4....2....4....1....2....5....4....4....2....6....2....6....6....6
%e A212539 ..3....6....8....4....8....5....8....5....5....6....7....6....2....6....6....8
%K A212539 nonn,new
%O A212539 1,2
%A A212539 _R. H. Hardin_ May 20 2012

%I A212538
%S A212538 1,4,8,15,21,30,33,46,53,66,67,87,88,95,111,125,126,143,144,170,180,
%T A212538 183,184,214,220,223,231,245,246,282,283,297,302,305,315,346,347,350,
%U A212538 355,388,389,412,413,420,442,445,446,478,481,495,499,507,508,526,533,553,557,560
%N A212538 Number of nondecreasing sequences of 5 1..n integers with every element dividing the sequence sum
%C A212538 Row 5 of A212536
%H A212538 R. H. Hardin, <a href="/A212538/b212538.txt">Table of n, a(n) for n = 1..210</a>
%e A212538 Some solutions for n=8
%e A212538 ..1....1....1....2....8....1....2....2....1....1....5....1....1....2....3....1
%e A212538 ..2....1....1....2....8....3....4....4....4....3....5....1....1....2....3....3
%e A212538 ..2....3....2....6....8....3....4....6....5....4....5....1....1....2....6....6
%e A212538 ..2....3....4....6....8....3....6....6....5....8....5....3....3....6....6....6
%e A212538 ..7....4....8....8....8....5....8....6....5....8....5....3....6....6....6....8
%K A212538 nonn,new
%O A212538 1,2
%A A212538 _R. H. Hardin_ May 20 2012

%I A212537
%S A212537 1,3,5,10,12,17,18,23,26,30,31,40,41,43,47,52,53,59,60,67,70,72,73,82,
%T A212537 84,86,89,94,95,103,104,109,111,113,115,125,126,128,130,137,138,144,
%U A212537 145,150,155,157,158,167,168,172,174,179,180,186,188,193,195,197,198,210,211,213
%N A212537 Number of nondecreasing sequences of 4 1..n integers with every element dividing the sequence sum
%C A212537 Row 4 of A212536
%H A212537 R. H. Hardin, <a href="/A212537/b212537.txt">Table of n, a(n) for n = 1..210</a>
%F A212537 Empirical: a(n) = 2*a(n-1) -2*a(n-2) -a(n-3) +4*a(n-4) -5*a(n-5) +6*a(n-7) -9*a(n-8) +3*a(n-9) +6*a(n-10) -12*a(n-11) +7*a(n-12) +4*a(n-13) -13*a(n-14) +11*a(n-15) -11*a(n-17) +13*a(n-18) -4*a(n-19) -7*a(n-20) +13*a(n-21) -8*a(n-22) -a(n-23) +10*a(n-24) -10*a(n-25) +5*a(n-26) +5*a(n-27) -10*a(n-28) +10*a(n-29) -a(n-30) -8*a(n-31) +13*a(n-32) -7*a(n-33) -4*a(n-34) +13*a(n-35) -11*a(n-36) +11*a(n-38) -13*a(n-39) +4*a(n-40) +7*a(n-41) -12*a(n-42) +6*a(n-43) +3*a(n-44) -9*a(n-45) +6*a(n-46) -5*a(n-48) +4*a(n-49) -a(n-50) -2*a(n-51) +2*a(n-52) -a(n-53)
%e A212537 Some solutions for n=8
%e A212537 ..4....2....6....7....5....2....1....3....2....1....3....2....4....1....1....1
%e A212537 ..4....6....6....7....5....2....2....3....2....2....3....3....6....1....1....1
%e A212537 ..8....8....6....7....5....4....3....6....4....2....3....3....6....1....2....4
%e A212537 ..8....8....6....7....5....8....6....6....4....5....3....4....8....3....4....6
%K A212537 nonn,new
%O A212537 1,2
%A A212537 _R. H. Hardin_ May 20 2012

%I A212536
%S A212536 1,2,1,3,2,1,4,3,3,1,5,4,5,3,1,6,5,7,5,4,1,7,6,8,10,8,4,1,8,7,11,12,
%T A212536 15,8,5,1,9,8,12,17,21,15,12,5,1,10,9,14,18,30,21,24,12,6,1,11,10,16,
%U A212536 23,33,40,33,29,16,6,1,12,11,18,26,46,44,69,40,39,16,7,1,13,12,19,30,53,64,83,91
%N A212536 T(n,k)=Number of nondecreasing sequences of n 1..k integers with every element dividing the sequence sum
%C A212536 Table starts
%C A212536 .1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
%C A212536 .1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
%C A212536 .1.3..5..7..8..11..12..14..16..18..19...22...23...25...27...29...30...33...34
%C A212536 .1.3..5.10.12..17..18..23..26..30..31...40...41...43...47...52...53...59...60
%C A212536 .1.4..8.15.21..30..33..46..53..66..67...87...88...95..111..125..126..143..144
%C A212536 .1.4..8.15.21..40..44..64..76.103.104..148..149..165..197..229..230..271..272
%C A212536 .1.5.12.24.33..69..83.116.145.188.193..290..293..332..428..496..497..606..607
%C A212536 .1.5.12.29.40..91.106.161.202.266.272..474..478..561..747..874..876.1141.1142
%C A212536 .1.6.16.39.57.130.157.245.331.439.455..867..878.1034.1417.1646.1651.2236.2240
%C A212536 .1.6.16.45.70.166.200.334.451.644.665.1424.1440.1713.2384.2785.2793.3927.3932
%H A212536 R. H. Hardin, <a href="/A212536/b212536.txt">Table of n, a(n) for n = 1..877</a>
%e A212536 Some solutions for n=8 k=4
%e A212536 ..1....1....2....1....1....1....1....2....1....1....1....1....2....2....1....1
%e A212536 ..1....2....2....2....1....1....1....3....1....1....1....1....2....2....2....1
%e A212536 ..2....3....3....2....1....1....1....3....1....2....1....2....2....4....2....2
%e A212536 ..2....3....3....2....1....1....1....3....1....2....1....2....2....4....3....2
%e A212536 ..2....3....3....2....1....2....2....3....1....2....1....2....4....4....4....3
%e A212536 ..4....4....3....3....2....2....2....3....1....2....1....2....4....4....4....3
%e A212536 ..4....4....4....3....2....4....2....3....2....2....2....2....4....4....4....3
%e A212536 ..4....4....4....3....3....4....2....4....2....2....4....4....4....4....4....3
%Y A212536 Column 3 is A000212(floor((n+5)/2))
%Y A212536 Row 3 is A106252
%K A212536 nonn,tabl,new
%O A212536 1,2
%A A212536 _R. H. Hardin_ May 20 2012

%I A212535
%S A212535 7,7,12,18,33,44,83,106,157,200,272,351,518,715,1010,1343,1756,2152,
%T A212535 2670,3101,3685,4168,4866,5474,6452,7345,8718,10082,11964,13801,16207,
%U A212535 18489,21386,24289,27876,31630,36294,41204,47131,53471,60792,68552,77355,86516
%N A212535 Number of nondecreasing sequences of n 1..7 integers with every element dividing the sequence sum
%C A212535 Column 7 of A212536
%H A212535 R. H. Hardin, <a href="/A212535/b212535.txt">Table of n, a(n) for n = 1..210</a>
%e A212535 Some solutions for n=8
%e A212535 ..3....1....1....1....1....1....1....1....1....1....1....1....2....6....2....2
%e A212535 ..3....1....1....1....1....1....1....1....2....3....1....1....2....6....2....2
%e A212535 ..3....3....1....1....2....1....1....1....2....3....2....3....4....6....2....2
%e A212535 ..3....3....1....5....2....1....3....1....2....3....2....3....4....6....2....3
%e A212535 ..3....4....1....5....2....1....3....1....2....3....3....3....4....6....2....3
%e A212535 ..5....4....5....5....2....1....3....1....2....3....3....3....4....6....4....4
%e A212535 ..5....4....5....6....5....3....6....2....4....4....6....4....4....6....4....4
%e A212535 ..5....4....5....6....5....3....6....4....5....4....6....6....4....6....6....4
%K A212535 nonn,new
%O A212535 1,1
%A A212535 _R. H. Hardin_ May 20 2012

%I A212534
%S A212534 6,6,11,17,30,40,69,91,130,166,224,296,439,606,841,1080,1352,1594,
%T A212534 1877,2112,2397,2672,3055,3500,4159,4932,5966,7144,8568,10073,11781,
%U A212534 13488,15367,17256,19348,21511,23999,26623,29660,32913,36620,40561,45024,49719
%N A212534 Number of nondecreasing sequences of n 1..6 integers with every element dividing the sequence sum
%C A212534 Column 6 of A212536
%H A212534 R. H. Hardin, <a href="/A212534/b212534.txt">Table of n, a(n) for n = 1..210</a>
%F A212534 Empirical: a(n) = a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15) +a(n-60) -a(n-61) -a(n-62) +a(n-65) +a(n-66) +a(n-67) -a(n-68) -a(n-69) -a(n-70) +a(n-73) +a(n-74) -a(n-75)
%e A212534 Some solutions for n=8
%e A212534 ..1....2....3....2....1....2....1....2....1....2....1....2....2....1....1....1
%e A212534 ..1....2....3....2....1....2....3....2....1....4....1....2....2....1....1....1
%e A212534 ..1....3....3....2....1....3....3....2....1....4....2....2....2....2....2....2
%e A212534 ..3....3....3....2....1....3....3....2....1....4....3....6....2....2....3....2
%e A212534 ..4....3....3....2....2....3....5....2....1....4....3....6....3....2....5....4
%e A212534 ..4....3....3....2....4....5....5....2....2....6....4....6....3....4....6....4
%e A212534 ..4....4....3....6....5....6....5....3....2....6....4....6....4....4....6....4
%e A212534 ..6....4....3....6....5....6....5....3....3....6....6....6....6....4....6....6
%K A212534 nonn,new
%O A212534 1,1
%A A212534 _R. H. Hardin_ May 20 2012

%I A212533
%S A212533 5,5,8,12,21,21,33,40,57,70,90,101,132,153,208,262,343,401,491,546,
%T A212533 625,667,737,770,851,889,989,1070,1226,1361,1592,1787,2070,2305,2616,
%U A212533 2864,3198,3444,3781,4045,4399,4670,5070,5391,5860,6254,6786,7235,7843,8336
%N A212533 Number of nondecreasing sequences of n 1..5 integers with every element dividing the sequence sum
%C A212533 Column 5 of A212536
%H A212533 R. H. Hardin, <a href="/A212533/b212533.txt">Table of n, a(n) for n = 1..210</a>
%F A212533 Empirical: a(n) = a(n-1) +a(n-2) -2*a(n-5) +a(n-8) +a(n-9) -a(n-10) +a(n-60) -a(n-61) -a(n-62) +2*a(n-65) -a(n-68) -a(n-69) +a(n-70)
%e A212533 Some solutions for n=8
%e A212533 ..1....1....5....1....1....1....1....2....1....1....1....1....1....1....2....1
%e A212533 ..1....1....5....2....1....2....2....2....1....1....1....3....1....1....3....1
%e A212533 ..1....2....5....2....2....2....3....3....1....2....1....3....1....3....3....1
%e A212533 ..1....4....5....3....2....5....3....3....3....2....1....3....1....5....3....1
%e A212533 ..4....4....5....4....2....5....3....5....3....2....1....5....2....5....3....2
%e A212533 ..4....4....5....4....2....5....4....5....3....4....1....5....2....5....3....2
%e A212533 ..4....4....5....4....2....5....4....5....3....4....1....5....4....5....3....2
%e A212533 ..4....4....5....4....4....5....4....5....3....4....1....5....4....5....4....2
%K A212533 nonn,new
%O A212533 1,1
%A A212533 _R. H. Hardin_ May 20 2012

%I A212532
%S A212532 4,4,7,10,15,15,24,29,39,45,57,65,83,92,111,127,149,163,193,213,245,
%T A212532 270,305,333,378,408,455,496,547,587,650,697,763,819,889,949,1033,
%U A212532 1096,1183,1261,1353,1431,1539,1625,1737,1836,1953,2057,2192,2300,2439,2566,2711
%N A212532 Number of nondecreasing sequences of n 1..4 integers with every element dividing the sequence sum
%C A212532 Column 4 of A212536
%H A212532 R. H. Hardin, <a href="/A212532/b212532.txt">Table of n, a(n) for n = 1..210</a>
%F A212532 Empirical: a(n) = a(n-1) +a(n-2) -a(n-4) -a(n-5) +a(n-6) +a(n-12) -a(n-13) -a(n-14) +a(n-16) +a(n-17) -a(n-18)
%e A212532 Some solutions for n=8
%e A212532 ..1....4....2....2....1....1....1....2....1....1....1....2....3....1....1....1
%e A212532 ..1....4....2....3....1....1....1....2....1....2....1....2....3....1....1....3
%e A212532 ..2....4....2....3....1....2....1....4....1....2....3....2....3....1....2....3
%e A212532 ..4....4....2....3....3....2....1....4....1....2....3....2....3....1....2....3
%e A212532 ..4....4....4....3....3....2....1....4....1....2....4....2....3....1....2....3
%e A212532 ..4....4....4....3....3....2....1....4....1....3....4....2....3....1....4....3
%e A212532 ..4....4....4....3....3....2....2....4....2....3....4....4....3....3....4....4
%e A212532 ..4....4....4....4....3....4....2....4....4....3....4....4....3....3....4....4
%K A212532 nonn,new
%O A212532 1,1
%A A212532 _R. H. Hardin_ May 20 2012

%I A212531
%S A212531 1,2,5,10,21,40,83,161,331,644,1102,3627,5646,10083,29188,53681,84121,
%T A212531 237453,373114,1400757,2443494
%N A212531 Number of nondecreasing sequences of n 1..n integers with every element dividing the sequence sum
%C A212531 Diagonal of A212536
%e A212531 Some solutions for n=8
%e A212531 ..1....2....2....2....1....1....1....1....2....1....1....1....4....6....1....2
%e A212531 ..1....2....2....2....1....1....4....1....2....1....1....2....7....6....2....2
%e A212531 ..2....2....2....2....4....2....4....1....4....1....1....2....7....6....3....2
%e A212531 ..2....2....3....2....5....4....5....3....4....2....1....2....7....6....3....3
%e A212531 ..4....2....5....2....5....4....5....3....4....3....2....3....7....6....3....3
%e A212531 ..4....2....5....4....8....4....5....3....8....4....2....4....8....6....3....4
%e A212531 ..7....4....5....7....8....4....8....6....8....4....2....4....8....6....3....4
%e A212531 ..7....4....6....7....8....4....8....6....8....8....2....6....8....6....6....4
%K A212531 nonn,new
%O A212531 1,2
%A A212531 _R. H. Hardin_ May 20 2012

%I A212090
%S A212090 0,1,16,80,251,610,1261,2331,3970,6351,9670,14146,20021,27560,37051,
%T A212090 48805,63156,80461,101100,125476,154015,187166,225401,269215,319126,
%U A212090 375675,439426,510966,590905,679876,778535,887561,1007656,1139545
%N A212090 Number of (w,x,y,z) with all terms in {1,...,n} and w<x+y+z.
%C A212090 a(n)+A000332(n+1) = n^4. For a guide to related sequences, see A211795.
%F A212090 a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).
%F A212090 G.f.: -x*(1+11*x+10*x^2+x^3) / (x-1)^5.
%F A212090 a(n) = n*(-2+(1+(2+23*n)*n)*n)/24.
%t A212090 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212090 (Do[If[w < x + y + z, s = s + 1],
%t A212090 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212090 Map[t[#] &, Range[0, 50]] (* A212090 *)
%t A212090 FindLinearRecurrence[%]
%t A212090 (* Peter Moses, Apr 13 2012 *)
%Y A212090 Cf. A211795, A212091.
%K A212090 nonn,easy,new
%O A212090 0,3
%A A212090 _Clark Kimberling_, May 01 2012

%I A212498
%S A212498 7,1,2,3,4,5,6,7,1,2,3,4,5,7,6,1,2,3,4,7,5,6,1,2,3,7,4,5,6,1,2,7,3,4,
%T A212498 5,6,1,2,7
%N A212498 A finite sequence (of length 39) in which every permutation of [1..7] is a substring.
%D A212498 Adleman, Leonard. Short permutation strings. Discrete Math. 10(1974), 197--200. MR0373911 (51 #10111)
%Y A212498 Cf. A212497.
%K A212498 nonn,fini,full,new
%O A212498 1,1
%A A212498 _N. J. A. Sloane_, May 19 2012

%I A212497
%S A212497 4,1,2,3,4,1,2,4,3,1,2,4
%N A212497 A finite sequence (of length 12) in which every permutation of [1..4] is a substring.
%D A212497 Adleman, Leonard. Short permutation strings. Discrete Math. 10(1974), 197--200. MR0373911 (51 #10111)
%Y A212497 Cf. A212498.
%K A212497 nonn,fini,full,new
%O A212497 1,1
%A A212497 _N. J. A. Sloane_, May 19 2012

%I A212496
%S A212496 1,2,1,0,1,2,3,2,1,2,3,2,3,4,3,4,5,4,5,4,3,4,5,6,5,6,7,6,7,6,7,6,5,
%T A212496 6,5,6,7,8,7,8,9,8,9,8,9,10,11,10,9,8,7,6,7,8,7,8,7,8,9,10
%V A212496 -1,-2,-1,0,1,2,3,2,1,2,3,2,3,4,3,4,5,4,5,4,3,4,5,6,5,6,7,6,7,6,7,6,5,
%W A212496 6,5,6,7,8,7,8,9,8,9,8,9,10,11,10,9,8,7,6,7,8,7,8,7,8,9,10
%N A212496 a(n) = Sum_{k=1..n} (-1)^{k-Omega(k)} with Omega(k) the total number of prime factors of k (counted with multiplicity).
%C A212496 On May 16, 2012 Zhi-Wei Sun conjectured that a(n) is positive for each n>4. He has verified this for n up to 10^{10}, and shown that the conjecture implies the famous Riemann Hypothesis. Moreover, he guessed that a(n)>sqrt(n) for any n>324 (and also a(n)<sqrt(n)log(log(n)) for n>5892); this implies that the sequence contains all natural numbers.
%C A212496 Sun also conjectured that b(n)=sum_{k=1}^n(-1)^{k-Omega(k)}/k<0 for all n=1,2,3,..., and verified this for n up to 2*10^9. Moreover, he guessed that b(n)<-1/sqrt(n) for all n>1, and b(n)>-log(log(n))/sqrt(n) for n>2008.
%H A212496 Zhi-Wei Sun, <a href="/A212496/b212496.txt">Table of n, a(n) for n = 1..100000</a>
%H A212496 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1204.6689">On a pair of zeta functions</a>, preprint, arxiv:1204.6689.
%H A212496 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b3990068.1205">On the parities of Omega(n)-n</a>, a message to Number Theory List, May 18, 2012.
%e A212496 We have a(4)=0 since (-1)^{1-Omega(1)}+(-1)^{2-Omega(2)}+(-1)^{3-Omega(3)}+(-1)^{4-Omega(4)}=-1-1+1+1=0.
%t A212496 PrimeDivisor[n_]:=Part[Transpose[FactorInteger[n]],1]
%t A212496 Omega[n_]:=If[n==1,0,Sum[IntegerExponent[n,Part[PrimeDivisor[n],i]],{i,1,Length[PrimeDivisor[n]]}]]
%t A212496 s[0]=0
%t A212496 s[n_]:=s[n]=s[n-1]+(-1)^(n-Omega[n])
%t A212496 Do[Print[n," ",s[n]],{n,1,100000}]
%Y A212496 Cf. A008836, A002819.
%K A212496 sign,nice,new
%O A212496 1,2
%A A212496 _Zhi-Wei Sun_, May 19 2012

%I A185022
%S A185022 5,7,17,19,29,47,59,89,127,139,167,199,227,239,257,269,397,409,419,
%T A185022 467,479,607,619,727,797,929,997,1009,1039,1277,1279,1427,1447,1459,
%U A185022 1487,1499,1559,1597,1697,1709,1777,1877,1889,1987,2087,2129,2269,2399,2609
%N A185022 Prime p such that p, p+12, p+24 are all primes.
%C A185022 Intersection of A046133 and A033560. - _M. F. Hasler_, May 19 2012
%H A185022 Salvatore Di Guida, <a href="/A185022/b185022.txt">Table of n, a(n) for n = 1..1000</a>
%o A185022 (PARI) forprime(p=1,2999,isprime(p+12)&isprime(p+24)&print1(p",")) \\ _M. F. Hasler_, May 19 2012
%K A185022 nonn,easy,new
%O A185022 1,1
%A A185022 _Salvatore Di Guida_, May 19 2012

%I A212091
%S A212091 0,0,0,3,3,3,6,12,12,24,24,33,36,42,48,63,63,72,84,99,99,132,141,159,
%T A212091 162,174,180,219,225,243,258,282,282,330,339,369,381,405,420,465,465,
%U A212091 492,525,558,567,627,645,681,684,732,744,804,810,846,885,930
%N A212091 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.
%C A212091 Every term is divisible by 3.  For a guide to related sequences, see A211795.
%t A212091 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212091 (Do[If[w^2 == x^2 + y^2 + z^2, s = s + 1],
%t A212091 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212091 Map[t[#] &, Range[0, 50]] (* A212091 *)
%t A212091 %/3  (* integers *)
%t A212091 (* Peter Moses, Apr 13 2012 *)
%Y A212091 Cf. A211795.
%K A212091 nonn,new
%O A212091 0,4
%A A212091 _Clark Kimberling_, May 02 2012

%I A210945
%S A210945 1,2,3,5,1,7,1,11,3,1,15,3,1,22,6,3,1,30,7,4,1,42,11,7,3,1,56,13,9,4,
%T A210945 1,77,20,15,8,3,1,101,23,18,10,4,1,135,33,27,17,8,3,1,176,40,34,22,11,
%U A210945 4,1,231,54,47,33,18,8,3,1,297,65,58,42,24,11,4,1
%N A210945 Triangle read by rows: T(n,k) = number of parts in the k-th column of the mirror of the last shell of the partitions of n.
%C A210945 For another version see A207379.
%H A210945 Alois P. Heinz, <a href="/A210945/b210945.txt">Rows n = 1..70</a>
%e A210945 For n = 7 the illustration shows two arrangements of the last shell of the partitions of 7:
%e A210945 .
%e A210945 .       (7)        (7)
%e A210945 .     (4+3)        (3+4)
%e A210945 .     (5+2)        (2+5)
%e A210945 .   (3+2+2)        (2+2+3)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .       (1)        (1)
%e A210945 .                 --------
%e A210945 .                  15,3,1
%e A210945 .
%e A210945 We can see that in the right hand picture (the mirror) the number of part for columns 1..3 are 15, 3, 1 therefore row 7 lists 15, 3, 1.
%e A210945 Written as a triangle begins:
%e A210945 1;
%e A210945 2;
%e A210945 3;
%e A210945 5,    1;
%e A210945 7,    1;
%e A210945 11,   3,  1;
%e A210945 15,   3,  1;
%e A210945 22,   6,  3,  1;
%e A210945 30,   7,  4,  1;
%e A210945 42,  11,  7,  3,  1;
%e A210945 56,  13,  9,  4,  1;
%e A210945 77,  20, 15,  8,  3,  1;
%e A210945 101, 23, 18, 10,  4,  1;
%e A210945 135, 33, 27, 17,  8,  3,  1;
%e A210945 176, 40, 34, 22, 11,  4,  1;
%e A210945 231, 54, 47, 33, 18,  8,  3,  1;
%e A210945 297, 65, 58, 42, 24, 11,  4,  1;
%Y A210945 Column 1 is A000041,n >= 1. Column 2 is A083751. Column 3 is A119907. Row sums give A138137.
%Y A210945 Cf. A135010, A138121, A182717, A207379.
%K A210945 nonn,tabf,new
%O A210945 1,2
%A A210945 _Omar E. Pol_, Apr 21 2012
%E A210945 More terms from _Alois P. Heinz_, May 07 2012

%I A212278
%S A212278 0,0,0,1,0,2,1,0,3,2,1,1,0,4,3,2,2,1,2,1,0,5,4,3,3,2,3,2,1,3,2,1,1,0,
%T A212278 6,5,4,4,3,4,3,2,4,3,2,2,1,4,3,2,2,1,2,1,0,7,6,5,5,4,5,4,3,5,4,3,3,2,
%U A212278 5,4,3,3,2,3,2,1,5,4,3,3,2,3,2,1,3,2,1,1,0,8
%N A212278 Number of adjacent pairs of zeros (possibly overlapping) in the representation of n in base of Fibonacci numbers (A014417).
%C A212278 a(n) = 0 only if n = Fibonacci(k)-1.
%H A212278 Alois P. Heinz, <a href="/A212278/b212278.txt">Table of n, a(n) for n = 0..10946</a>
%e A212278 A014417(5) = 1000, two pairs of adjacent zeros, so a(5) = 2.
%p A212278 F:= combinat[fibonacci]:
%p A212278 b:= proc(n) option remember; local j;
%p A212278       if n=0 then 0
%p A212278     else for j from 2 while F(j+1)<=n do od;
%p A212278          b(n-F(j))+2^(j-2)
%p A212278       fi
%p A212278     end:
%p A212278 a:= proc(n) local c, h, m, t;
%p A212278       c, t, m:= 0, 1, b(n);
%p A212278       while m>0 do
%p A212278         h:= irem(m, 2, 'm');
%p A212278         if h=t and h=0 then c:=c+1 fi;
%p A212278         t:=h
%p A212278       od; c
%p A212278     end:
%p A212278 seq (a(n), n=0..150);  # _Alois P. Heinz_, May 18 2012
%Y A212278 Cf. A000045, A003714, A014417, A007895, A102364.
%K A212278 base,nonn,new
%O A212278 0,6
%A A212278 _Alex Ratushnyak_, May 13 2012

%I A211774
%S A211774 0,0,0,3,12,60,420,3255,28056,270144,2868840,33293205,419329020,
%T A211774 5697423732,83069039508,1293734268645,21436030749840,376516868504160,
%U A211774 6988441065717744,136675039085498691,2809247116432575420,60543293881318183740,1365186080156105513460
%N A211774 Number of rooted 2-regular labelled graphs on n nodes.
%H A211774 Alois P. Heinz, <a href="/A211774/b211774.txt">Table of n, a(n) for n = 0..170</a>
%F A211774 a(n) = n*A001205(n).
%F A211774 E.g.f.: x*A'(x) where A(x) = exp(-x/2-x^2/4)/sqrt(1-x) is the e.g.f. for A001205.
%p A211774 egf:= x *diff (exp (-x/2-x^2/4)/sqrt(1-x), x):
%p A211774 a:= n-> n! * coeff (series (egf, x, n+1), x, n):
%p A211774 seq (a(n), n=0..30);  # _Alois P. Heinz_, May 18 2012
%t A211774 nn = 20; a = Log[1/(1 - x)]/2 - x/2 - x^2/4; Drop[Range[0, nn]! CoefficientList[Series[x D[Exp[a], x], {x, 0, nn}], x], 3]
%K A211774 nonn,new
%O A211774 0,4
%A A211774 _Geoffrey Critzer_, May 18 2012

%I A212089
%S A212089 0,1,9,45,139,333,684,1258,2133,3402,5167,7542,10656,14647,19665,
%T A212089 25875,33451,42579,53460,66304,81333,98784,118903,141948,168192,
%U A212089 197917,231417,269001,310987,357705,409500,466726,529749,598950,674719
%N A212089 Number of (w,x,y,z) with all terms in {1,...,n} and w>=average{x,y,z}.
%C A212089 Also, number of (w,x,y,z) with all terms in {1,...,n} and w<=average{x,y,z}.  A212089(n)+A212088(n)=n^4.  For a guide to related sequences, see A211795.
%H A212089 <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F A212089 a(n) = 4a(n-1)-6a(n-2)+5a(n-3)-5a(n-4)+6a(n-5)-4a(n-6)+a(n-7).
%F A212089 G.f.: x*(1+7*x^4+8*x^3+15*x^2+5*x) / ((x^2+x+1)*(-x+1)^5).
%t A212089 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212089 (Do[If[3 w >= x + y + z, s = s + 1],
%t A212089 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212089 Map[t[#] &, Range[0, 50]] (* A212088 *)
%t A212089 FindLinearRecurrence[%]
%t A212089 (* Peter Moses, Apr 13 2012 *)
%Y A212089 Cf. A211795, A212069, A212088.
%K A212089 nonn,new
%O A212089 0,3
%A A212089 _Clark Kimberling_, May 01 2012

%I A212088
%S A212088 0,0,7,36,117,292,612,1143,1963,3159,4833,7099,10080,13914,18751,
%T A212088 24750,32085,40942,51516,64017,78667,95697,115353,137893,163584,
%U A212088 192708,225559,262440,303669,349576,400500,456795,518827,586971,661617
%N A212088 Number of (w,x,y,z) with all terms in {1,...,n} and w<average{x,y,z}.
%C A212088 Also, number of (w,x,y,z) with all terms in {1,...,n} and w>average{x,y,z}.  A212088(n) + A212089(n) = n^4. For a guide to related sequences, see A211795.
%H A212088 <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (4,-6,5,-5,6,-4,1).
%F A212088 a(n) = 4a(n-1)-6a(n-2)+5a(n-3)-5a(n-4)+6a(n-5)-4a(n-6)+a(n-7).
%F A212088 G.f.: x^2*(x^4+5*x^3+15*x^2+8*x+7) / ((x^2+x+1)*(1-x)^5).
%t A212088 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212088 (Do[If[3 w < x + y + z, s = s + 1],
%t A212088 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212088 Map[t[#] &, Range[0, 50]] (* A212088 *)
%t A212088 FindLinearRecurrence[%]
%t A212088 (* Peter Moses, Apr 13 2012 *)
%Y A212088 Cf. A211795, A212069, A212089.
%K A212088 nonn,new
%O A212088 0,3
%A A212088 _Clark Kimberling_, May 01 2012

%I A211773
%S A211773 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,
%T A211773 1,37,71,101,127,149,167,181,191,197,199,197,191,181,
%U A211773 167,149,127,101,71,37,1,43,89,139,193,251,313,379,449,523,601,683,769
%V A211773 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,
%W A211773 1,-37,-71,-101,-127,-149,-167,-181,-191,-197,-199,-197,-191,-181,
%X A211773 -167,-149,-127,-101,-71,-37,1,43,89,139,193,251,313,379,449,523,601,683,769
%N A211773 Prime-generating polynomial: 2*n^2 - 108*n + 1259.
%C A211773 This polynomial generates 92 primes (66 distinct ones) for n from 0 to 99 (in fact the next two terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by the Euler's polynomial for n = m-35, which is m^2 - 69*m + 1231 (see the link below), but having six distinct primes more than this one.
%C A211773 The non-prime terms in the first 100 are: 1 (taken twice), 1369 = 37^2, 1849 = 43^2, 4033 = 37*109, 5633 = 43*131, 7739 = 71*109 and 8251 = 37*223.
%C A211773 For n = 2*m-34 we obtain the polynomial 8*m^2 - 488*m + 7243, which generates 31 primes in a row starting from m=0 (polynomial already reported, see the link below).
%C A211773 For n = 4*m-34 we obtain the polynomial 32*m^2 - 976*m + 7243, which generates 31 primes in row starting from m=0.
%D A211773 Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.
%H A211773 Bruno Berselli, <a href="/A211773/b211773.txt">Table of n, a(n) for n = 0..1000</a>
%H A211773 Joe L. Mott and Kermite Rose, <a href="http://webcache.googleusercontent.com/search?q=cache:TRI9WtS_BIQJ:www.math.fsu.edu/~aluffi/archive/paper134.ps.gz+%2295+primes%22+Boston&amp;cd=2&amp;hl=ro&amp;ct=clnk&amp;gl=ro">Prime-Producing Cubic Polynomials</a>
%H A211773 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial">MathWorld: Prime-Generating Polynomial</a>
%H A211773 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211773 G.f.: (1259-2624*x+1369*x^2)/(1-x)^3. [_Bruno Berselli_, May 18 2012]
%o A211773 (MAGMA) [2*n^2-108*n+1259: n in [0..49]]; // _Bruno Berselli_, May 18 2012
%K A211773 sign,easy,new
%O A211773 0,1
%A A211773 _Marius Coman_, May 18 2012

%I A212087
%S A212087 0,1,6,15,28,45,66,95,132,173,210,267,320,385,458,523,600,693,786,899,
%T A212087 1000,1109,1226,1367,1492,1629,1778,1931,2084,2269,2426,2615,2812,
%U A212087 3013,3222,3427,3624,3857,4094,4335,4564,4841,5082,5379,5656,5913
%N A212087 Number of (w,x,y,z) with all terms in {1,...,n} and w^2+x^2=y^2+z^2.
%C A212087 For a guide to related sequences, see A211795.
%t A212087 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212087 (Do[If[w^2 + x^2 == y^2 + z^2, s = s + 1],
%t A212087 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212087 Map[t[#] &, Range[0, 60]] (* A212087 *)
%t A212087 (* Peter Moses, Apr 13 2012 *)
%Y A212087 Cf. A211795.
%K A212087 nonn,new
%O A212087 0,3
%A A212087 _Clark Kimberling_, May 02 2012

%I A211775
%S A211775 5419,5209,5003,4801,4603,4409,4219,4033,3851,3673,3499,3329,3163,
%T A211775 3001,2843,2689,2539,2393,2251,2113,1979,1849,1723,1601,1483,1369,
%U A211775 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,1,37,71,101
%V A211775 5419,5209,5003,4801,4603,4409,4219,4033,3851,3673,3499,3329,3163,
%W A211775 3001,2843,2689,2539,2393,2251,2113,1979,1849,1723,1601,1483,1369,
%X A211775 1259,1153,1051,953,859,769,683,601,523,449,379,313,251,193,139,89,43,1,-37,-71,-101
%N A211775 Prime-generating polynomial: 2*n^2 - 212*n + 5419.
%C A211775 This polynomial generates 92 primes (57 distinct ones) for n from 0 to 99 (in fact the next seven terms are still primes but we keep the range 0-99, customary for comparisons), just three primes less than the record held by the Euler's polynomial for n = m-35, which is m^2 - 69*m + 1231 (see the link below).
%C A211775 The non-prime terms in the first 100 are: 1, 1369 = 37^2, 1849 = 43^2, 4033 = 37*109 (all taken twice).
%C A211775 For n = 2*m+54 we obtain the polynomial 8*m^2 + 8*m - 197, which generates 31 primes in a row starting from m = 0 (the polynomial 8*m^2 - 488*m + 7243 generates the same 31 primes, but in reverse order).
%D A211775 Joe L. Mott and Kermite Rose, Prime-Producing Cubic Polynomials in Lecture Notes in Pure and Applied Mathematics (Vol. 220), Marcel Dekker Inc., 2001, pages 281-317.
%H A211775 Bruno Berselli, <a href="/A211775/b211775.txt">Table of n, a(n) for n = 0..1000</a>
%H A211775 Joe L. Mott and Kermite Rose, <a href="http://webcache.googleusercontent.com/search?q=cache:TRI9WtS_BIQJ:www.math.fsu.edu/~aluffi/archive/paper134.ps.gz+%2295+primes%22+Boston&amp;cd=2&amp;hl=ro&amp;ct=clnk&amp;gl=ro"> Prime-Producing Cubic Polynomials</a>
%H A211775 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial">MathWorld: Prime-Generating Polynomial</a>
%H A211775 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A211775 G.f.: (5419-11048*x+5633*x^2)/(1-x)^3. [_Bruno Berselli_, May 18 2012]
%o A211775 (MAGMA) [2*n^2-212*n+5419: n in [0..49]]; // _Bruno Berselli_, May 18 2012
%K A211775 sign,easy,new
%O A211775 0,1
%A A211775 _Marius Coman_, May 18 2012

%I A212492
%S A212492 7,19,31,61,97,127,139,181,229,271,337,409,421,607,631,811,1009,1021,
%T A212492 1039,1051,1279,1291,1471,1597,1609,1657,1777,1861,1867,1987,2017,
%U A212492 2131,2371,2539,2647,2677,2719,2791,3109,3319,3361,3457,3517,3529,3547,3571,3907
%N A212492 Prime p such that p, p+10, p+12 are all primes.
%H A212492 Salvatore Di Guida, <a href="/A212492/b212492.txt">Table of n, a(n) for n = 1..85</a>
%t A212492 Select[Range[5000], PrimeQ[#] && PrimeQ[#+10] && PrimeQ[#+12] &] (* _T. D. Noe_, May 18 2012 *)
%K A212492 nonn,new
%O A212492 1,1
%A A212492 _Salvatore Di Guida_, May 18 2012

%I A210840
%S A210840 0,1,256,6561,65536,390625,1679616,5764801,16777216,43046721,1,2,257,
%T A210840 6562,65537,390626,1679617,5764802,16777217,43046722,256,257,512,6817,
%U A210840 65792,390881,1679872,5765057,16777472,43046977,6561,6562,6817,13122,72097,397186
%N A210840 Sum of the 8th powers of the digits of n.
%C A210840 This is to exponent 8 as A007953 is to exponent 0, A003132 is to exponent 2, and A055013 is to exponent 4. The subsequence of primes (for n = 11, 12, 14, 21, 41, ...) begins 2, 257, 65537, 65537.
%e A210840 a(12) = 1^8 + 2^8 = 257.
%t A210840 Table[Total[IntegerDigits[n]^8], {n, 0, 100}] (* _T. D. Noe_, May 18 2012 *)
%Y A210840 Cf. A007953, A003132, A055013.
%K A210840 nonn,base,easy,new
%O A210840 0,3
%A A210840 _Jonathan Vos Post_, May 10 2012

%I A212274
%S A212274 5,12,16,20,64,76,49,100,112,64,136,148,160,172,184,105,120,220,120,
%T A212274 244,256,121,280,292,161,316,144,176,352,364,221,217,400,217,424,436,
%U A212274 232,225,472,484,496,288,273,532,225,288,253,580,352,604,616,276,640
%N A212274 Minimal k >= 5n such that n^2 + 2nk + k is a perfect square.
%C A212274 Without any restriction, trivially, a(n)=0 and, if to consider a(n) positive, then, again trivially, a(n)=1. Without the restriction k >= 5n, we have a(n)=4*n+1; on the other hand, if to require a(n)>=5*n and in addition a(n+1)>a(n), then we obtain sequence 5,12,16,20 and, beginning with n=5, we have progression 64+12*(n-5).
%t A212274 Table[k = 5*n; While[! IntegerQ[Sqrt[n^2 + 2*n*k + k]], k++]; k, {n, 100}] (* _T. D. Noe_, May 18 2012 *)
%K A212274 nonn,new
%O A212274 1,1
%A A212274 _Vladimir Shevelev_ and Peter Moses, May 13 2012

%I A212491
%S A212491 1,1,2,10,109,2344,95526,7321508,1062894966,295821381776,
%T A212491 159750645147982,169055785061863986,353149667541498542122,
%U A212491 1463495730873823283553016,12070551778811708797824867546,198534598750957942596957876692320,6519958358404237723197281898833187776
%N A212491 G.f.: A(x) = x + x*ITERATE^2(x + x*ITERATE^4(x + x*ITERATE^8(x + x*ITERATE^16(x + ...)))), where ITERATE^n(F(x)) denotes the n-th iteration of F(x), and the nesting of iterations continue indefinitely.
%e A212491 G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 109*x^5 + 2344*x^6 + 95526*x^7 +...
%e A212491 where A(x) is generated by nesting 2^n-th iterations of shifted series:
%e A212491 A(x) = x + x*B(B(x));
%e A212491 B(x) = x + x*C(C(C(C(x))));
%e A212491 C(x) = x + x*D(D(D(D(D(D(D(D(x))))))));
%e A212491 D(x) = x + x*E(E(E(E(E(E(E(E(E(E(E(E(E(E(E(E(x)))))))))))); ...
%e A212491 The above series begin:
%e A212491 B(x) = x + x^2 + 4*x^3 + 44*x^4 + 1006*x^5 + 43392*x^6 + 3459214*x^7 +...
%e A212491 C(x) = x + x^2 + 8*x^3 + 184*x^4 + 8620*x^5 + 746176*x^6 + 117816292*x^7 +...
%e A212491 D(x) = x + x^2 + 16*x^3 + 752*x^4 + 71320*x^5 + 12374144*x^6 +...
%e A212491 E(x) = x + x^2 + 32*x^3 + 3040*x^4 + 580144*x^5 + 201551104*x^6 +...
%o A212491 PARI) /* Define the (2^n)-th iteration of function F: */
%o A212491 {ITERATE2(n, F, p=#F)=local(G=F); for(i=1, n, G=subst(G, x, G+x*O(x^p))); G}
%o A212491 /* A(x) results from nested iterations of shifted series: */
%o A212491 {a(n)=local(A=x); for(k=0, n, A=ITERATE2(n-k, x + x*A, n)); polcoeff(A, n)}
%o A212491 for(n=1,20,print1(a(n),","))
%Y A212491 Cf. A195192, A205320, A205319.
%K A212491 nonn,new
%O A212491 1,3
%A A212491 _Paul D. Hanna_, May 18 2012

%I A212396
%S A212396 0,0,3,23,41,313,73,676,3439,38231,46169,602359,703999,10565707,
%T A212396 12071497,13669093,30716561,582722017,215455199,4516351061,991731385,
%U A212396 361369795,393466951,9817955321,31848396101,858318957533,922672670033,8903430207697,9522990978097
%N A212396 Numerator of the average number of move operations required by an insertion sort of n (distinct) elements.
%C A212396 The average number of move operations is 1/n! times the number of move operations required to sort all permutations of [n] (A212395), assuming that each permutation is equiprobable.
%H A212396 Alois P. Heinz, <a href="/A212396/b212396.txt">Table of n, a(n) for n = 0..700</a>
%H A212396 Wikipedia, <a href="http://en.wikipedia.org/wiki/Insertion_sort">Insertion sort</a>
%F A212396 a(n) = numerator of A212395(n)/A000142(n).
%F A212396 a(n) = numerator of n*(n+7)/4 - 2*H(n) with n-th harmonic number H(n) = Sum_{k=1..n} 1/k = A001008(n)/A002805(n).
%F A212396 a(n) = numerator of n*(n+7)/4 - 2*(Psi(n+1)+gamma) with digamma function Psi and the Euler-Mascheroni constant gamma = A001620.
%e A212396 0/1, 0/1, 3/2, 23/6, 41/6, 313/30, 73/5, 676/35, 3439/140, 38231/1260, 46169/1260, 602359/13860, 703999/13860 ... = A212396/A212397
%p A212396 b:= proc(n) option remember;
%p A212396       `if`(n=0, 0, b(n-1)*n + (n-1)! * (n-1)*(n+4)/2)
%p A212396     end:
%p A212396 a:= n-> numer(b(n)/n!):
%p A212396 seq (a(n), n=0..30);
%p A212396 # second Maple program
%p A212396 a:= n-> numer (expand (n*(n+7)/4 -2*(Psi(n+1)+gamma))):
%p A212396 seq (a(n), n=0..30);
%Y A212396 Denominators are in A212397.
%Y A212396 Cf. A000142, A001008, A001620, A002805, A212395.
%K A212396 nonn,frac,new
%O A212396 0,3
%A A212396 _Alois P. Heinz_, May 14 2012

%I A212397
%S A212397 1,1,2,6,6,30,5,35,140,1260,1260,13860,13860,180180,180180,180180,
%T A212397 360360,6126120,2042040,38798760,7759752,2586584,2586584,59491432,
%U A212397 178474296,4461857400,4461857400,40156716600,40156716600,1164544781400,1164544781400
%N A212397 Denominator of the average number of move operations required by an insertion sort of n (distinct) elements.
%C A212397 The average number of move operations is 1/n! times the number of move operations required to sort all permutations of [n] (A212395), assuming that each permutation is equiprobable.
%H A212397 Alois P. Heinz, <a href="/A212397/b212397.txt">Table of n, a(n) for n = 0..700</a>
%H A212397 Wikipedia, <a href="http://en.wikipedia.org/wiki/Insertion_sort">Insertion sort</a>
%F A212397 a(n) = denominator of A212395(n)/A000142(n).
%p A212397 b:= proc(n) option remember;
%p A212397       `if`(n=0, 0, b(n-1)*n + (n-1)! * (n-1)*(n+4)/2)
%p A212397     end:
%p A212397 a:= n-> denom(b(n)/n!):
%p A212397 seq (a(n), n=0..30);
%Y A212397 Numerators are in A212396.
%Y A212397 Cf. A000142, A212395.
%K A212397 nonn,frac,new
%O A212397 0,3
%A A212397 _Alois P. Heinz_, May 14 2012

%I A212395
%S A212395 0,0,3,23,164,1252,10512,97344,990432,11010528,132966720,1734793920,
%T A212395 24330205440,365150833920,5840673108480,99204809356800,
%U A212395 1783428104908800,33833306484633600,675513065777356800,14160039606855475200,310935875030323200000
%N A212395 Number of move operations required to sort all permutations of [n] by insertion sort.
%C A212395 a(n) is n! times the average number of move operations (A212396, A212397) required by an insertion sort of n (distinct) elements.
%H A212395 Alois P. Heinz, <a href="/A212395/b212395.txt">Table of n, a(n) for n = 0..170</a>
%H A212395 Wikipedia, <a href="http://en.wikipedia.org/wiki/Insertion_sort">Insertion sort</a>
%F A212395 a(n) = a(n-1)*n + (n-1)! * (n-1)*(n+4)/2 for n>0, a(0) = 0.
%F A212395 a(n) = n! * (n*(n+7)/4 - 2*H(n)) with with n-th harmonic number H(n) = Sum_{k=1..n} 1/k = A001008(n)/A002805(n).
%e A212395 a(0) = a(1) = 0 because 0 or 1 elements are already sorted.
%e A212395 a(2) = 3: [1,2] is sorted and [2,1] needs 3 moves.
%e A212395 a(3) = 23: [1,2,3]->(0), [1,3,2]->(3), [2,1,3]->(3), [2,3,1]->(4), [3,1,2]->(6), [3,2,1]->(7); sum of all moves gives 0+3+3+4+6+7 = 23.
%p A212395 a:= proc(n) option remember;
%p A212395       `if`(n=0, 0, a(n-1)*n + (n-1)! * (n-1)*(n+4)/2)
%p A212395     end:
%p A212395 seq (a(n), n=0..30);
%Y A212395 Cf. A001008, A002805, A159324, A212396, A212397.
%K A212395 nonn,new
%O A212395 0,3
%A A212395 _Alois P. Heinz_, May 14 2012

%I A211113
%S A211113 2,0,7,8,8,6,2,2,4,9,7,7,3,5,4,5,6,6,0,1,7,3,0,6,7,2,5,3,9,7,0,4,9,3,
%T A211113 0,2,2,2,6,2,6,8,5,3,1,2,8,7,6,7,2,5,3,7,6,1,0,1,1,3,5,5,7,1,0,6,1,4,
%U A211113 7,2,9,1,9,3,2,2,9,2,3,4,0,4,8,7,5,4,3,2,6,6,9,4,0,7,3,3,2,1,5,6,4,3,1,0,9,9,7,5,6
%N A211113 Decimal expansion of -zeta(-1/2).
%C A211113 zeta(1/2) = -zeta(3/2)/(4*Pi); zeta(3/2) being A078434.
%H A211113 Wikipedia, <a href="http://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>
%e A211113 0.207886224977354566017306725397049302226...
%o A211113 (PARI) -zeta(-1/2) \\ _Charles R Greathouse IV_, May 18 2012
%Y A211113 Cf. A059750, A078434.
%K A211113 nonn,cons,easy,new
%O A211113 0,1
%A A211113 _Stanislav Sykora_, May 17 2012

%I A212479
%S A212479 5,6,7,5,5,5,1,6,3,3,0,6,9,5,7,8,2,5,3,8,4,6,1,3,1,4,4,1,9,2,4,5,3,3,
%T A212479 4,3,9,0,3,2,2,9,7,6,6,6,6,3,9,3,3,9,9,7,0,9,7,3,8,9,2,7,6,5,7,6,4,5,
%U A212479 9,5,6,7,4,5,9,7,7,3,0,6,5,9,8,8,6,0,8,4,8,7,7,5,9,9,2,9,9,5,1,6,6,3,9,7,8,5,6,7
%N A212479 Decimal expansion of the absolute value of infinite power tower of i.
%C A212479 This c = |z|, where z is the complex solution of z = i^z or, eqivalently, z = i^i^i^...
%H A212479 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%F A212479 c = |i^i^i^...|.
%e A212479 0.5675551633069578253846131441924533439 ...
%o A212479 PARI/GP: start with I^(0.4+0.4*I) and iterate (%+I^%)/2. It converges pretty rapidly to z.
%Y A212479 Cf. A077589 (real part of z), A077590 (imaginary part of z), A212480 (argument of z).
%K A212479 nonn,cons,easy,new
%O A212479 0,1
%A A212479 _Stanislav Sykora_, May 17 2012

%I A212480
%S A212480 6,8,8,4,5,3,2,2,7,1,0,7,7,0,2,1,3,0,4,9,8,7,6,7,5,7,1,1,7,6,8,2,4,2,
%T A212480 5,9,6,0,8,0,9,5,4,4,3,2,3,2,2,2,3,1,3,5,5,2,8,6,8,6,9,2,3,2,1,0,4,4,
%U A212480 9,7,0,7,3,0,1,9,4,0,3,2,7,4,3,8,3,5,2,5,7,3,1,1,0,2,3,0,1,6,5,8,9,7,0,3,0,8,1,5
%N A212480 Decimal expansion of the argument of infinite power tower of i.
%C A212480 This c, expressed in radiants, equals arg(z), where z is the complex solution of z = i^z or, equivalently, z = i^i^i^... Also, c = atan(A077590/A077589).
%H A212480 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%F A212480 c = arg(i^i^i^...).
%e A212480 0.6884532271077021304987675711768242596 ...
%o A212480 PARI/GP: start with I^(0.4+0.4*I) and iterate (%+I^%)/2. It converges pretty rapidly to z.
%Y A212480 Cf. A077589 (real part of z), A077590 (imaginary part of z), A212479 (absolute value of z).
%K A212480 nonn,cons,easy,new
%O A212480 0,1
%A A212480 _Stanislav Sykora_, May 17 2012

%I A212069
%S A212069 0,1,2,9,22,41,72,115,170,243,334,443,576,733,914,1125,1366,1637,1944,
%T A212069 2287,2666,3087,3550,4055,4608,5209,5858,6561,7318,8129,9000,9931,
%U A212069 10922,11979,13102,14291,15552,16885,18290,19773,21334,22973
%N A212069 Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z.
%C A212069 w is the average of {x,y,z}, as well as {w,x,y,z}.
%C A212069 For a guide to related sequences, see A211795.
%F A212069 a(n)=3a*n-1)-3a(n-2)+2a(n-3)-3a(n-4)+3a(n-5)-a(n-6).
%t A212069 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212069 (Do[If[3 w == x + y + z, s = s + 1],
%t A212069 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212069 Map[t[#] &, Range[0, 50]] (* A212087 *)
%t A212069 FindLinearRecurrence[%]
%t A212069 (* Peter Moses, Apr 13 2012 *)
%Y A212069 Cf. A211795.
%K A212069 nonn,new
%O A212069 0,3
%A A212069 _Clark Kimberling_, May 01 2012

%I A212068
%S A212068 0,0,3,10,25,49,86,137,206,294,405,540,703,895,1120,1379,1676,2012,
%T A212068 2391,2814,3285,3805,4378,5005,5690,6434,7241,8112,9051,10059,11140,
%U A212068 12295,13528,14840,16235,17714,19281,20937,22686,24529,26470,28510
%N A212068 Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.
%C A212068 For a guide to related sequences, see A211795.
%F A212068 a(n)=3a*n-1)-2a(n-2)-2a*n-3)+3a(n-4)-a(n-5).
%t A212068 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212068 (Do[If[2 w == x + y + z, s = s + 1],
%t A212068 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212068 Map[t[#] &, Range[0, 50]] (* A212086 *)
%t A212068 FindLinearRecurrence[%]
%t A212068 (* Peter Moses, Apr 13 2012 *)
%Y A212068 Cf. A211795.
%K A212068 nonn,new
%O A212068 0,3
%A A212068 _Clark Kimberling_, May 01 2012

%I A212067
%S A212067 0,1,2,3,10,11,12,13,26,45,46,47,60,61,62,63,88,89,120,121,128,129,
%T A212067 130,131,162,199,200,255,262,263,264,265,332,333,334,335,402,403,404,
%U A212067 405,436,437,438,439,446,477,478,479,540,601,674,675,682,683,786
%N A212067 Number of (w,x,y,z) with all terms in {1,...,n} and w^3=x*y*z.
%C A212067 For a guide to related sequences, see A211795.
%e A212067 a(4) counts these ten 4-tuples:
%e A212067 (1,1,1,1), (2,2,2,2), (3,3,3,3), (4,4,4,4),
%e A212067 (2,1,2,4), (2,1,4,2), (2,2,1,4), (2,2,4,1),
%e A212067 (2,4,1,2), (2,4,2,1).
%t A212067 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212067 (Do[If[w^3 == x*y*z, s = s + 1],
%t A212067 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212067 Map[t[#] &, Range[0, 60]] (* A212067 *)
%t A212067 (* Peter Moses, Apr 13 2012 *)
%Y A212067 Cf. A211795, A212068.
%K A212067 nonn,new
%O A212067 0,3
%A A212067 _Clark Kimberling_, Apr 30 2012

%I A210734
%S A210734 3,23,2467,246809,246811,24681012141619,
%T A210734 24681012141618202224262830323436384041,
%U A210734 24681012141618202224262830323436384042444649
%N A210734 Primes p such that p + 1 or p - 1 is a concatenation of successive even numbers starting from 2.
%C A210734 a(9) has 625 digits, a(10) has 1476 digits, a(11) has 5048 digits, a(12) has 39024 digits.
%H A210734 Arkadiusz Wesolowski, <a href="/A210734/b210734.txt">Table of n, a(n) for n = 1..10</a>
%H A210734 G. L. Honaker, Jr. and Chris Caldwell, <a href="http://primes.utm.edu/curios/cpage/20588.html">24681...50451 (625-digits)</a>
%H A210734 Henri & Renaud Lifchitz, <a href="http://www.primenumbers.net/prptop/searchform.php?form=%3F24681012141618202224%3F&amp;action=Search">PRP Records</a>
%t A210734 lst = {}; c = 0; Do[c = c*10^IntegerLength[n] + n; a = c - 1; If[PrimeQ[a], AppendTo[lst, a]]; b = c + 1; If[PrimeQ[b], AppendTo[lst, b]], {n, 2, 48, 2}]; lst
%Y A210734 Cf. A019520.
%K A210734 base,nonn,new
%O A210734 1,1
%A A210734 _Arkadiusz Wesolowski_, May 10 2012

%I A182459
%S A182459 1,2,13,20,46,157,236,532,1198,4045,6068,13654,46084,103690,1181101,
%T A182459 1771652,3986218,102162424,229865455,344798183,517197275,775795913,
%U A182459 1163693870,3927466813,5891200220,13255200496,29824201117,44736301676,100656678772,226477527238
%N A182459 Numbers n of initial person such that the n-th person survives in the duck-duck-goose game.
%C A182459 In more detail: n students are sitting in a circle. A professor starts tagging them in the pattern - duck, duck, goose, ... . If a student is tagged goose he or she leaves the circle immediately. The last remaining student is the winner. These are the numbers n of initial students such that the n-th student will be the winner.
%H A182459 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JosephusProblem.html">Josephus Problem</a>
%H A182459 Wikipedia, <a href="http://en.wikipedia.org/wiki/Josephus_problem">Josephus problem</a>
%F A182459 a(n) = A081615(n)-1.
%K A182459 nonn,new
%O A182459 1,2
%A A182459 _Dan Fodor_, Apr 30 2012

%I A211487
%S A211487 0,1,1,1,1,1,1,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,1,0,1,1,1,0,1,0,1,0,0,1,
%T A211487 0,0,1,1,0,0,1,0,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0,1,0,
%U A211487 0,0,1,0,1,1,0,0,0,0,1,0,1,1,1,0,0,1,0,0
%N A211487 Characteristic sequence of numbers n having a primitive root modulo n.
%C A211487 a(1)=0, since we have an empty set of numbers more than 0 and less than 1.
%C A211487 If A(x) is the counting function of a(n)=1, n<=x, then A(x)~2*x/ln(x) as x tends to infinity.
%F A211487 a(n)=1 iff n=2,4, p^k, 2*p^k, where p is an odd prime.
%F A211487 A001783(n)==(-1)^a(n) mod n.
%Y A211487 Cf. A001783.
%K A211487 nonn,new
%O A211487 1
%A A211487 _Vladimir Shevelev_, May 13 2012

%I A212478
%S A212478 5864,172980,3708268,70600212,1246724695,21292987433,357117150362,
%T A212478 5923063781205,97514964737576,1597131825489558,26058934302628094,
%U A212478 423954039651724411,6881835973423757103,111509561797692820078
%N A212478 Number of 0..3 arrays of length 2*n+6 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212478 Row 7 of A212471
%H A212478 R. H. Hardin, <a href="/A212478/b212478.txt">Table of n, a(n) for n = 1..210</a>
%e A212478 Some solutions for n=3
%e A212478 ..0....2....2....0....2....0....0....2....0....0....0....0....0....0....2....2
%e A212478 ..2....0....2....2....2....2....0....0....0....2....0....2....0....0....2....2
%e A212478 ..0....2....0....0....0....2....0....0....2....0....0....2....2....2....0....0
%e A212478 ..0....2....0....0....2....0....0....2....2....0....2....0....0....0....2....0
%e A212478 ..2....2....0....2....0....0....0....0....2....0....2....2....0....2....2....2
%e A212478 ..0....0....0....0....2....2....0....0....0....0....0....2....0....0....0....2
%e A212478 ..0....2....0....0....0....0....2....0....0....3....0....0....1....1....2....0
%e A212478 ..0....0....3....0....3....0....0....3....1....2....2....1....1....3....2....2
%e A212478 ..3....1....3....3....2....3....0....1....0....0....2....0....3....0....1....0
%e A212478 ..1....2....2....1....0....1....0....2....0....1....0....2....2....2....1....3
%e A212478 ..3....2....0....1....0....1....0....0....3....0....0....2....0....1....1....0
%e A212478 ..2....2....0....2....3....1....3....1....0....0....2....3....0....1....2....3
%K A212478 nonn,new
%O A212478 1,1
%A A212478 _R. H. Hardin_ May 17 2012

%I A212477
%S A212477 2037,53553,1098260,19960518,344305566,5795798788,96239314549,
%T A212477 1584499716270,25938577854660,422900154613448,6874291237701437,
%U A212477 111487268294929274,1804868382278013414,29177249831017201248
%N A212477 Number of 0..3 arrays of length 2*n+5 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212477 Row 6 of A212471
%H A212477 R. H. Hardin, <a href="/A212477/b212477.txt">Table of n, a(n) for n = 1..210</a>
%e A212477 Some solutions for n=3
%e A212477 ..2....0....2....0....2....0....0....2....2....2....2....0....0....0....2....0
%e A212477 ..2....2....0....2....0....0....0....0....0....0....0....0....0....2....0....0
%e A212477 ..0....0....0....2....0....2....0....0....0....2....0....0....2....2....0....2
%e A212477 ..0....2....2....2....2....0....2....2....0....0....2....2....0....2....2....2
%e A212477 ..1....1....0....3....2....1....1....3....0....3....2....2....3....1....1....0
%e A212477 ..2....1....2....0....1....1....2....2....0....0....0....0....0....0....3....0
%e A212477 ..2....2....0....0....2....2....1....0....1....0....2....1....0....0....1....1
%e A212477 ..2....1....2....2....0....1....0....2....1....1....1....3....2....3....0....1
%e A212477 ..0....1....1....1....0....1....2....0....2....1....0....0....3....1....2....0
%e A212477 ..2....1....1....2....3....3....0....0....2....1....2....1....1....2....0....0
%e A212477 ..1....3....2....1....2....1....1....1....0....1....0....1....3....2....1....3
%K A212477 nonn,new
%O A212477 1,1
%A A212477 _R. H. Hardin_ May 17 2012

%I A212476
%S A212476 707,16649,321320,5622580,95010466,1578372444,25966120647,
%T A212476 424537002094,6911873593516,112193588288892,1817051683302947,
%U A212476 29377851622547612,474335448720482746,7650319463541214660,123278749437509026298
%N A212476 Number of 0..3 arrays of length 2*n+4 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212476 Row 5 of A212471
%H A212476 R. H. Hardin, <a href="/A212476/b212476.txt">Table of n, a(n) for n = 1..210</a>
%e A212476 Some solutions for n=3
%e A212476 ..2....0....2....2....0....0....0....0....2....0....2....2....0....2....0....0
%e A212476 ..0....2....0....2....2....2....0....2....0....0....0....2....0....2....0....2
%e A212476 ..0....3....1....3....3....2....3....3....2....2....3....0....1....1....0....2
%e A212476 ..3....3....3....2....1....3....1....0....0....2....2....2....2....1....3....0
%e A212476 ..1....0....0....0....0....0....2....1....0....2....0....3....2....0....1....0
%e A212476 ..0....0....0....0....0....0....0....1....1....1....0....0....3....0....1....1
%e A212476 ..0....0....1....0....2....1....0....0....2....0....1....0....0....0....1....1
%e A212476 ..0....3....3....2....1....2....3....3....0....0....2....0....1....2....3....3
%e A212476 ..0....2....1....2....1....3....1....2....0....0....2....3....1....2....0....3
%e A212476 ..1....1....1....3....3....0....3....2....1....0....1....3....1....1....1....0
%K A212476 nonn,new
%O A212476 1,1
%A A212476 _R. H. Hardin_ May 17 2012

%I A212475
%S A212475 246,5211,93308,1581686,26242268,430682205,7023308036,114062278982,
%T A212475 1847205725100,29854017540652,481755564098784,7764949351300178,
%U A212475 125039067197089316,2011992600375317822,32354843213417739992
%N A212475 Number of 0..3 arrays of length 2*n+3 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212475 Row 4 of A212471
%H A212475 R. H. Hardin, <a href="/A212475/b212475.txt">Table of n, a(n) for n = 1..210</a>
%e A212475 Some solutions for n=3
%e A212475 ..1....2....1....1....3....2....3....2....2....0....0....1....1....1....1....0
%e A212475 ..1....0....0....0....1....1....2....1....1....0....1....1....1....0....1....0
%e A212475 ..0....0....2....1....0....0....3....1....0....0....2....3....0....0....1....2
%e A212475 ..0....0....3....2....2....3....1....0....0....1....1....0....0....2....1....2
%e A212475 ..2....1....0....3....0....0....0....3....0....1....1....2....3....0....3....0
%e A212475 ..0....0....1....0....2....0....0....1....3....2....0....0....1....0....0....1
%e A212475 ..3....2....1....0....0....2....0....1....1....1....0....0....0....1....1....0
%e A212475 ..0....0....0....0....1....3....3....2....1....0....2....2....2....2....1....3
%e A212475 ..3....3....3....3....1....0....2....1....1....0....3....1....2....2....3....3
%K A212475 nonn,new
%O A212475 1,1
%A A212475 _R. H. Hardin_ May 17 2012

%I A212474
%S A212474 85,1597,27024,445780,7274268,118029501,1908601444,30794657082,
%T A212474 496095308012,7983014938756,128351434757356,2062286003105664,
%U A212474 33118464621596452,531627920798750598,8530886270887938488
%N A212474 Number of 0..3 arrays of length 2*n+2 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212474 Row 3 of A212471
%H A212474 R. H. Hardin, <a href="/A212474/b212474.txt">Table of n, a(n) for n = 1..210</a>
%e A212474 Some solutions for n=3
%e A212474 ..0....0....1....2....2....0....2....2....1....2....1....1....1....2....0....3
%e A212474 ..0....0....2....0....0....0....2....0....0....1....1....0....0....1....1....0
%e A212474 ..1....2....1....1....1....1....1....3....2....0....0....1....0....2....3....2
%e A212474 ..0....1....0....1....3....2....0....0....2....0....0....0....3....0....0....0
%e A212474 ..2....1....0....0....2....1....0....1....2....0....1....0....0....0....0....1
%e A212474 ..2....0....1....1....1....0....3....1....0....1....2....2....0....0....0....1
%e A212474 ..1....2....3....3....2....3....3....2....0....2....3....1....3....0....3....2
%e A212474 ..1....2....1....2....0....2....0....0....1....2....3....2....1....3....0....2
%K A212474 nonn,new
%O A212474 1,1
%A A212474 _R. H. Hardin_ May 17 2012

%I A212473
%S A212473 30,486,7862,126606,2034200,32644314,523487828,8390738062,
%T A212473 134446993320,2153764911576,34495886876214,552428007496418,
%U A212473 8845775486728392,141630619844515292,2267494906950636872,36300174404152871822
%N A212473 Number of 0..3 arrays of length 2*n+1 with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212473 Row 2 of A212471
%H A212473 R. H. Hardin, <a href="/A212473/b212473.txt">Table of n, a(n) for n = 1..210</a>
%e A212473 Some solutions for n=3
%e A212473 ..0....2....2....2....0....1....0....1....0....0....2....2....1....3....0....0
%e A212473 ..0....0....0....0....0....1....3....1....0....1....2....0....0....0....1....1
%e A212473 ..2....3....2....1....2....0....0....0....1....1....3....0....1....0....1....3
%e A212473 ..1....3....3....0....0....3....1....0....1....3....0....1....0....0....0....0
%e A212473 ..2....0....1....3....1....2....2....1....1....0....0....1....1....1....0....3
%e A212473 ..0....0....1....1....2....2....1....3....2....0....1....0....0....2....1....0
%e A212473 ..2....0....2....1....2....1....2....0....1....0....1....0....0....0....2....2
%K A212473 nonn,new
%O A212473 1,1
%A A212473 _R. H. Hardin_ May 17 2012

%I A212472
%S A212472 10,150,2338,36814,582440,9240426,146861788,2337014158,37222215640,
%T A212472 593246686360,9460081927746,150915638082466,2408347433725048,
%U A212472 38443582673708348,613802451466325528,9802073528030277518
%N A212472 Number of 0..3 arrays of length 2*n with sum no more than 3*n in any length 2n subsequence (=50% duty cycle)
%C A212472 Row 1 of A212471
%H A212472 R. H. Hardin, <a href="/A212472/b212472.txt">Table of n, a(n) for n = 1..210</a>
%e A212472 Some solutions for n=3
%e A212472 ..0....1....2....1....3....1....0....0....1....3....3....1....3....2....1....0
%e A212472 ..3....3....1....0....3....0....1....3....2....0....1....1....3....0....1....3
%e A212472 ..1....2....0....3....1....0....2....0....0....3....1....3....0....0....1....2
%e A212472 ..2....2....3....1....0....3....0....3....0....1....1....0....2....1....2....0
%e A212472 ..0....0....0....1....0....2....1....1....2....0....3....1....0....3....3....0
%e A212472 ..3....0....1....1....2....2....0....2....1....0....0....2....1....3....0....0
%K A212472 nonn,new
%O A212472 1,1
%A A212472 _R. H. Hardin_ May 17 2012

%I A212471
%S A212471 10,150,30,2338,486,85,36814,7862,1597,246,582440,126606,27024,5211,
%T A212471 707,9240426,2034200,445780,93308,16649,2037,146861788,32644314,
%U A212471 7274268,1581686,321320,53553,5864,2337014158,523487828,118029501,26242268
%N A212471 T(n,k)=Number of 0..3 arrays of length n+2*k-1 with sum no more than 3*k in any length 2k subsequence (=50% duty cycle)
%C A212471 Table starts
%C A212471 ....10....150.....2338.....36814.....582440.....9240426.....146861788
%C A212471 ....30....486.....7862....126606....2034200....32644314.....523487828
%C A212471 ....85...1597....27024....445780....7274268...118029501....1908601444
%C A212471 ...246...5211....93308...1581686...26242268...430682205....7023308036
%C A212471 ...707..16649...321320...5622580...95010466..1578372444...25966120647
%C A212471 ..2037..53553..1098260..19960518..344305566..5795798788...96239314549
%C A212471 ..5864.172980..3708268..70600212.1246724695.21292987433..357117150362
%C A212471 .16886.558743.12564894.248263590.4504766041.78186208521.1325550644016
%H A212471 R. H. Hardin, <a href="/A212471/b212471.txt">Table of n, a(n) for n = 1..936</a>
%e A212471 Some solutions for n=3 k=4
%e A212471 ..0....2....2....2....2....2....2....2....0....2....0....0....0....0....0....2
%e A212471 ..0....2....0....2....2....2....0....0....0....2....0....0....2....0....0....0
%e A212471 ..0....2....1....2....2....0....0....0....2....1....1....0....0....1....0....3
%e A212471 ..3....2....0....2....0....0....3....0....0....0....0....0....0....0....2....0
%e A212471 ..1....0....0....1....0....2....2....1....2....2....0....3....0....0....1....2
%e A212471 ..0....0....3....1....3....2....0....3....1....1....0....1....3....0....3....3
%e A212471 ..0....0....1....0....0....1....2....0....1....0....0....0....1....0....3....0
%e A212471 ..3....0....3....2....3....0....1....0....0....3....0....2....2....0....2....0
%e A212471 ..2....1....2....2....0....2....1....0....2....1....3....0....3....2....0....2
%e A212471 ..0....0....0....1....1....2....0....1....1....0....2....3....3....0....0....0
%Y A212471 Column 1 is A006357(n+1)
%K A212471 nonn,tabl,new
%O A212471 1,1
%A A212471 _R. H. Hardin_ May 17 2012

%I A212470
%S A212470 146861788,523487828,1908601444,7023308036,25966120647,96239314549,
%T A212470 357117150362,1325550644016,4918335298665,18232047941771,
%U A212470 67489103954016,249349516426188,919083218908202,3377977784235134,12373106402582424
%N A212470 Number of 0..3 arrays of length n+13 with sum no more than 21 in any length 14 subsequence (=50% duty cycle)
%C A212470 Column 7 of A212471
%H A212470 R. H. Hardin, <a href="/A212470/b212470.txt">Table of n, a(n) for n = 1..36</a>
%e A212470 Some solutions for n=3
%e A212470 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212470 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212470 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212470 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212470 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212470 ..2....2....0....0....2....0....0....0....2....0....2....2....0....0....0....2
%e A212470 ..0....0....2....2....0....0....2....2....0....2....0....0....2....2....0....0
%e A212470 ..2....2....0....2....0....2....2....2....0....2....2....0....2....2....0....2
%e A212470 ..2....0....2....2....0....0....0....2....2....0....0....2....0....2....0....2
%e A212470 ..2....0....0....0....0....0....2....2....0....0....2....2....2....0....0....0
%e A212470 ..2....3....3....2....0....2....1....2....0....3....1....1....1....3....2....1
%e A212470 ..1....1....0....2....1....3....1....2....0....3....0....3....3....3....1....3
%e A212470 ..0....3....2....2....3....1....0....0....1....3....3....0....2....3....2....2
%e A212470 ..3....0....3....0....3....2....3....1....0....1....1....3....1....3....0....2
%e A212470 ..1....2....0....1....1....2....0....0....0....1....2....2....1....2....2....2
%e A212470 ..3....0....2....3....0....1....1....2....2....0....1....1....0....0....2....0
%K A212470 nonn,new
%O A212470 1,1
%A A212470 _R. H. Hardin_ May 17 2012

%I A212469
%S A212469 9240426,32644314,118029501,430682205,1578372444,5795798788,
%T A212469 21292987433,78186208521,286703599009,1049119428341,3828209603876,
%U A212469 13919545400124,50392127733527,182633247168566,663071942478215,2410713609407184
%N A212469 Number of 0..3 arrays of length n+11 with sum no more than 18 in any length 12 subsequence (=50% duty cycle)
%C A212469 Column 6 of A212471
%H A212469 R. H. Hardin, <a href="/A212469/b212469.txt">Table of n, a(n) for n = 1..37</a>
%e A212469 Some solutions for n=3
%e A212469 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212469 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A212469 ..2....0....0....0....2....0....0....2....0....0....0....2....0....2....0....0
%e A212469 ..2....0....0....0....0....0....2....2....2....2....2....0....2....0....0....2
%e A212469 ..0....0....2....0....2....0....2....0....0....2....2....2....0....0....0....0
%e A212469 ..0....0....0....2....2....0....2....0....2....0....0....2....2....2....2....0
%e A212469 ..0....0....0....2....2....0....0....0....2....2....2....2....2....2....2....2
%e A212469 ..0....0....2....0....2....0....2....2....0....2....0....0....0....2....0....2
%e A212469 ..2....2....0....0....2....0....0....0....2....2....2....2....2....0....2....2
%e A212469 ..3....0....3....3....0....2....2....2....1....2....1....2....0....1....0....1
%e A212469 ..1....0....0....2....1....0....2....3....3....3....3....1....3....2....3....0
%e A212469 ..0....2....1....3....2....0....2....1....0....0....3....2....0....3....3....3
%e A212469 ..1....1....1....0....1....2....0....1....2....1....1....3....0....0....0....2
%e A212469 ..0....2....2....1....2....0....2....3....3....1....0....0....1....3....3....3
%K A212469 nonn,new
%O A212469 1,1
%A A212469 _R. H. Hardin_ May 17 2012

%I A212468
%S A212468 582440,2034200,7274268,26242268,95010466,344305566,1246724695,
%T A212468 4504766041,16223101320,58162431112,207319630999,740101423056,
%U A212468 2647902413670,9488805055639,34037887492235,122166508976307
%N A212468 Number of 0..3 arrays of length n+9 with sum no more than 15 in any length 10 subsequence (=50% duty cycle)
%C A212468 Column 5 of A212471
%H A212468 R. H. Hardin, <a href="/A212468/b212468.txt">Table of n, a(n) for n = 1..56</a>
%e A212468 Some solutions for n=3
%e A212468 ..2....0....0....2....2....2....0....0....2....2....0....2....0....2....0....2
%e A212468 ..0....0....2....0....0....2....2....0....0....0....0....2....0....0....0....0
%e A212468 ..0....0....0....2....0....2....2....0....0....0....2....0....0....0....0....2
%e A212468 ..0....0....0....0....0....2....0....0....0....2....2....2....2....0....2....2
%e A212468 ..0....2....0....2....2....0....0....0....0....2....2....0....0....2....0....0
%e A212468 ..0....2....0....0....2....2....2....0....0....0....0....0....2....0....0....0
%e A212468 ..0....0....2....0....2....2....2....2....0....2....0....0....2....0....2....0
%e A212468 ..2....0....2....3....2....0....0....0....3....1....3....1....1....0....3....3
%e A212468 ..3....3....2....2....1....1....2....2....3....0....0....3....2....3....0....1
%e A212468 ..2....3....2....1....3....1....2....2....1....2....3....0....1....0....1....3
%e A212468 ..2....3....3....2....1....1....0....2....3....1....0....1....2....2....1....0
%e A212468 ..2....2....1....1....0....3....3....0....1....1....1....3....3....3....1....2
%K A212468 nonn,new
%O A212468 1,1
%A A212468 _R. H. Hardin_ May 17 2012

%I A212467
%S A212467 36814,126606,445780,1581686,5622580,19960518,70600212,248263590,
%T A212467 865983608,3027296124,10612842491,37272366925,131018316920,
%U A212467 460663150544,1619367566685,5689883208343,19982014129382,70158722229804
%N A212467 Number of 0..3 arrays of length n+7 with sum no more than 12 in any length 8 subsequence (=50% duty cycle)
%C A212467 Column 4 of A212471
%H A212467 R. H. Hardin, <a href="/A212467/b212467.txt">Table of n, a(n) for n = 1..210</a>
%e A212467 Some solutions for n=3
%e A212467 ..2....2....0....0....0....2....0....0....2....0....0....0....0....2....0....2
%e A212467 ..2....2....0....2....2....2....2....0....0....2....2....0....0....0....0....2
%e A212467 ..2....0....1....2....0....2....0....0....2....0....0....0....0....3....0....0
%e A212467 ..3....2....0....1....2....0....0....0....3....3....0....3....3....0....1....0
%e A212467 ..0....3....2....0....1....0....1....2....0....0....2....3....0....2....1....0
%e A212467 ..0....1....1....0....1....1....2....2....2....3....1....0....0....1....2....1
%e A212467 ..0....1....2....2....1....1....3....3....0....0....2....1....1....0....3....3
%e A212467 ..2....0....2....1....2....2....2....3....2....3....0....0....3....2....1....3
%e A212467 ..1....1....2....2....0....2....1....0....2....0....2....1....0....0....3....1
%e A212467 ..3....3....1....3....1....1....0....0....0....2....2....3....1....1....1....0
%K A212467 nonn,new
%O A212467 1,1
%A A212467 _R. H. Hardin_ May 17 2012

%I A212466
%S A212466 2338,7862,27024,93308,321320,1098260,3708268,12564894,42730092,
%T A212466 145559900,495992448,1689131389,5747566618,19552052026,66524165343,
%U A212466 226393294930,770536024747,2622515229619,8925194014628,30373724176979
%N A212466 Number of 0..3 arrays of length n+5 with sum no more than 9 in any length 6 subsequence (=50% duty cycle)
%C A212466 Column 3 of A212471
%H A212466 R. H. Hardin, <a href="/A212466/b212466.txt">Table of n, a(n) for n = 1..210</a>
%e A212466 Some solutions for n=3
%e A212466 ..1....1....1....3....2....0....0....0....3....2....1....3....3....1....2....2
%e A212466 ..2....2....2....0....0....0....3....0....0....2....1....0....3....1....1....1
%e A212466 ..0....0....0....0....2....2....3....3....1....1....2....0....3....2....3....3
%e A212466 ..2....2....1....0....1....1....1....1....0....0....3....3....0....0....3....1
%e A212466 ..2....2....3....1....2....2....0....0....1....0....2....1....0....0....0....1
%e A212466 ..0....2....1....0....0....0....0....1....0....3....0....0....0....3....0....0
%e A212466 ..0....0....1....2....1....2....2....0....3....2....0....2....0....0....0....0
%e A212466 ..0....0....3....3....1....0....2....2....1....1....0....0....0....0....3....1
%K A212466 nonn,new
%O A212466 1,1
%A A212466 _R. H. Hardin_ May 17 2012

%I A212465
%S A212465 150,486,1597,5211,16649,53553,172980,558743,1801720,5808944,18737526,
%T A212465 60447764,194984049,628912080,2028584905,6543438398,21106550023,
%U A212465 68080857417,219600230083,708339701336,2284812500323,7369860423165
%N A212465 Number of 0..3 arrays of length n+3 with sum no more than 6 in any length 4 subsequence (=50% duty cycle)
%C A212465 Column 2 of A212471
%H A212465 R. H. Hardin, <a href="/A212465/b212465.txt">Table of n, a(n) for n = 1..210</a>
%F A212465 Empirical: a(n) = 2*a(n-1) +3*a(n-2) +17*a(n-4) -8*a(n-5) -42*a(n-6) -8*a(n-7) -59*a(n-8) +26*a(n-9) +147*a(n-10) +33*a(n-11) +112*a(n-12) -115*a(n-13) -293*a(n-14) -5*a(n-15) -93*a(n-16) +221*a(n-17) +370*a(n-18) -65*a(n-19) +3*a(n-20) -242*a(n-21) -279*a(n-22) +78*a(n-23) +21*a(n-24) +176*a(n-25) +184*a(n-26) -33*a(n-27) -15*a(n-28) -87*a(n-29) -132*a(n-30) -7*a(n-31) +36*a(n-32) +35*a(n-33) +69*a(n-34) +5*a(n-35) -30*a(n-36) -10*a(n-37) -20*a(n-38) +10*a(n-40) +a(n-41) +2*a(n-42) -a(n-44)
%e A212465 Some solutions for n=3
%e A212465 ..3....0....1....1....0....0....0....1....1....0....0....3....0....0....2....1
%e A212465 ..1....3....1....0....3....2....0....3....2....2....3....1....0....3....0....1
%e A212465 ..1....1....1....2....0....0....0....1....0....2....0....0....0....0....0....0
%e A212465 ..1....0....2....1....0....0....0....0....1....0....2....2....2....1....2....3
%e A212465 ..0....2....1....1....3....3....3....2....0....2....0....1....1....0....2....1
%e A212465 ..2....2....2....1....0....2....0....3....2....2....2....1....2....2....2....2
%K A212465 nonn,new
%O A212465 1,1
%A A212465 _R. H. Hardin_ May 17 2012

%I A212263
%S A212263 1,1,2,5,16,58,222,869,3438,13672,54518,217706,870036,3478446,
%T A212263 13910128,55632657,222513784,890019102,3559999490,14239834188,
%U A212263 56958988812,227835217794,911339311462,3645353954182,14581408883620
%N A212263 Main diagonal of symmetric array defined by the recurrence T(n,1)=1, T(1,k)=1, for n >= k: T(n,k) = Sum_{i=1 to k-1} T(n-i,k), for n < k: T(n,k) = Sum_{i=1 to n-1} T(k-i,n).
%C A212263 a(n) mod 2 = A023900(n) mod 2. The recurrence mentioned in the title is the same as the recurrence in array A191898 but without the minus signs.
%F A212263 Main diagonal of array defined by: T(n,1)=1, T(1,k)=1, n>=k: Sum from i=1 to k-1 of T(n-i,k), n<k: Sum from i=1 to n-1 of T(k-i,n).
%t A212263 Clear[nn, t, n, k]; nn = 25;t[n_, 1] = 1;t[1, k_] = 1;t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k], {i, 1, k - 1}], Sum[t[k - i, n], {i, 1, n - 1}]];Table[t[n, n], {n, 1, nn}]
%Y A212263 Cf. A023900.
%K A212263 nonn,new
%O A212263 1,3
%A A212263 _Mats Granvik_, May 12 2012

%I A212410
%S A212410 1,0,1,8,7,6,5,4,11,3,10,2,9,1,8,4,7,3,6,2,5,1,13,4,8,12,3,7,11,2,6,
%T A212410 10,1,5,9,13,4,8,12,3,7,11,2,6,10,1,5,9,35,4,8,34,3,7,33,2,6,38,32,1,
%U A212410 5,37,31,9,4,36,30,8,3,35,29,7,2,34,28,6,1,33,27
%N A212410 Let f(n) = n + floor(log(n)). Then a(n) is the smallest number of iterations of f on n such that the perfect square is obtained, or 0 if no such square exists.
%C A212410 a(2) = 0 because f(f(f(….f(2)))…) = 2.
%C A212410 a(n)=1 for n = {1, 3, 14, 22, 33, 46, 60, 77, 96, …}
%H A212410 Michel Lagneau, <a href="/A212410/b212410.txt">Table of n, a(n) for n = 1..10000</a>
%e A212410 a(5) = 7 because:
%e A212410 f(5)=6;
%e A212410 f(f(5))=7;
%e A212410 f(f(f(5)))=8;
%e A212410 f(f(f(f(5))))=10;
%e A212410 f(f(f(f(f(5)))))=12;
%e A212410 f(f(f(f(f(f(5))))))=14;
%e A212410 f(f(f(f(f(f(f(5)))))))=16.
%e A212410 The first square number in this sequence 6,7,8,10,12,14,16 is on the seventh place and therefore a(5)=7.
%p A212410 with(numtheory): for n from 1 to 100 do:n0:=n:i:=0:for k from 1 to 1000 while(i=0) do:n1:=n0+floor(log(n0)): n2:=sqrt(n1):if n2 = floor(n2) then printf(`%d, `,k):i:=1:else n0:=n1:fi:od:if i=0 then printf(`%d, `,0):else fi:od:
%t A212410 f[n_] := Length@NestWhileList[#+Floor[Log[#]]&, n, !IntegerQ[Sqrt[#]] || #==n&] - 1; Table[f[n], {n, 3, 100}]
%K A212410 nonn,new
%O A212410 1,4
%A A212410 _Michel Lagneau_, May 15 2012

%I A212279
%S A212279 0,0,0,28,17,39,4,72,79,65,17,65,17,29,145,65,84,65,145,17,109,17,65,
%T A212279 0,145,65,17,145,88,17,64,145,17,28,257,65,17,65,145,145,257,65,17,
%U A212279 269,145,401,257,145,65,257,65,145,17,577,145,65,145,17,577,65,577
%N A212279 A002144(n+1)^2+1 mod A002144(n), where A002144 are the Pythagorean primes (p=4k+1).
%C A212279 Motivated by the fact that the first terms are zero (which is of course a coincidence). Other values (17, 65, 145, 257...) occur much more frequently.
%C A212279 Conjecture: a(n) = A082073(n)^2 + 1 for all n > 159. - _Charles R Greathouse IV_, May 13 2012
%H A212279 K. Rose, <a href="http://tech.groups.yahoo.com/group/primenumbers/message/24241">Law of small numbers</a>, primenumbers group, May 2012
%e A212279 5^2+1 = 2*13, 13^2+1 = 10*17, 17^2=10*29; therefore a(1)=a(2)=a(3)=0.
%e A212279 29^2+1 = 22*37+28, therefore a(4)=28.
%o A212279 (PARI) o=5;forprime(p=o+1,900,p%4==1|next;print1((o^2+1)%o=p","))
%K A212279 nonn,new
%O A212279 1,4
%A A212279 _M. F. Hasler_, May 13 2012

%I A212444
%S A212444 0,1,3,6,13,27,54,108,216,433,867,1734,3469,6939,13878,27756,55512,
%T A212444 111025,222050,444101,888202,1776404,3552808,7105617,14211235,
%U A212444 28422470,56844941,113689883,227379766,454759532,909519064,1819038129,3638076259,7276152518
%N A212444 Iterates A212439, starting from 0.
%C A212444 2*a(n) <= a(n+1) <= 2*a(n) + 1.
%H A212444 Reinhard Zumkeller, <a href="/A212444/b212444.txt">Table of n, a(n) for n = 0..100</a>
%H A212444 Benjamin Chaffin and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/CNC.pdf">The Curling Number Conjecture</a>, preprint.
%F A212444 a(n+1) = A212439(a(n)) = 2*a(n) + A181935(a(n)) mod 2, a(0) = 0.
%o A212444 (Haskell)
%o A212444 a212444 n = a212444_list !! n
%o A212444 a212444_list = iterate a212439 0
%K A212444 nonn,new
%O A212444 0,3
%A A212444 _Reinhard Zumkeller_, May 17 2012

%I A212441
%S A212441 0,1,2,5,6,7,8,9,13,14,17,18,22,23,24,25,29,30,31,32,33,34,38,39,40,
%T A212441 41,42,46,49,50,55,56,57,61,62,65,66,70,71,72,77,78,81,85,86,87,88,89,
%U A212441 93,94,95,96,97,98,102,103,104,105,106,110,113,114,119,120
%N A212441 Numbers with odd curling numbers of their binary representations, cf. A181935.
%C A212441 A212412(a(n)) = 1; complement of A212440.
%H A212441 Reinhard Zumkeller, <a href="/A212441/b212441.txt">Table of n, a(n) for n = 1..10000</a>
%o A212441 (Haskell)
%o A212441 a212441 n = a212441_list !! (n-1)
%o A212441 a212441_list = filter (odd . a181935) [0..]
%K A212441 nonn,new
%O A212441 1,3
%A A212441 _Reinhard Zumkeller_, May 17 2012

%I A212440
%S A212440 3,4,10,11,12,15,16,19,20,21,26,27,28,35,36,37,43,44,45,47,48,51,52,
%T A212440 53,54,58,59,60,63,64,67,68,69,73,74,75,76,79,80,82,83,84,90,91,92,99,
%U A212440 100,101,107,108,109,111,112,115,116,117,118,122,123,124,131
%N A212440 Numbers with even curling numbers of their binary representations, cf. A181935.
%C A212440 A212412(a(n)) = 0; complement of A212441.
%H A212440 Reinhard Zumkeller, <a href="/A212440/b212440.txt">Table of n, a(n) for n = 1..10000</a>
%o A212440 (Haskell)
%o A212440 a212440 n = a212440_list !! (n-1)
%o A212440 a212440_list = filter (even . a181935) [0..]
%K A212440 nonn,new
%O A212440 1,1
%A A212440 _Reinhard Zumkeller_, May 17 2012

%I A212439
%S A212439 1,3,5,6,8,11,13,15,17,19,20,22,24,27,29,30,32,35,37,38,40,42,45,47,
%T A212439 49,51,52,54,56,59,61,63,65,67,69,70,72,74,77,79,81,83,85,86,88,90,93,
%U A212439 94,96,99,101,102,104,106,108,111,113,115,116,118,120,123,125
%N A212439 2 * n + A181935(n) mod 2.
%C A212439 a(n) = 2*n + A212412(n): binary representation of n appended by the parity of its curling number;
%C A212439 A212444 gives iterations starting from 0.
%H A212439 Reinhard Zumkeller, <a href="/A212439/b212439.txt">Table of n, a(n) for n = 0..8191</a>
%H A212439 Benjamin Chaffin and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/CNC.pdf">The Curling Number Conjecture</a>, preprint.
%H A212439 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%o A212439 (Haskell)
%o A212439 a212439 n = 2 * n + a212412 n
%Y A212439  Cf. A014601, A007088.
%K A212439 nonn,new
%O A212439 0,2
%A A212439 _Reinhard Zumkeller_, May 17 2012

%I A212412
%S A212412 1,1,1,0,0,1,1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0,1,1,1,1,0,0,0,1,1,1,1,1,
%T A212412 1,0,0,0,1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,
%U A212412 0,0,1,1,1,0,0,0,0,1,1,0,0,1,0,0,0,1
%N A212412 Parity of curling number of binary expansion of n.
%C A212412 a(n) = A181935(n) mod 2;
%C A212412 a(A212440(n) = 0 and a(A212441(n) = 1;
%C A212412 A212439(n) = 2*n + a(n).
%H A212412 Reinhard Zumkeller, <a href="/A212412/b212412.txt">Table of n, a(n) for n = 0..10000</a>
%o A212412 (Haskell)
%o A212412 a212412 = (`mod` 2) . a181935
%Y A212412 Cf. A212444.
%K A212412 nonn,new
%O A212412 0
%A A212412 _Reinhard Zumkeller_, May 17 2012

%I A212442
%S A212442 1,8,140,1864,26602,373080,5253564,73911192,1040045475,14634444720,
%T A212442 205922568360,2897549559600,40771618763540,573700205699920,
%U A212442 8072574516567400,113589743388536528,1598328982089075749,22490195492277648120,316461065874934143252
%N A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.
%C A212442 More generally, exp(Sum_{k>=1} A002203(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A002203(2*k+1)*x - x^2)^binomial(2*n+1,n-k).
%C A212442 Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).
%F A212442 G.f.: 1 / ( (1+2*x-x^2)^3 * (1-14*x-x^2) ).
%F A212442 G.f.: 1 / Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) where A212443(n) = (1/n)*Sum_{d|n} moebius(n/d)*A002203(d)^2.
%e A212442 G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 +...
%e A212442 where
%e A212442 log(A(x)) = 2^3*x + 6^3*x^2/2 + 14^3*x^3/3 + 34^3*x^4/4 + 82^3*x^5/5 + 198^3*x^6/6 + 478^3*x^7/7 + 1154^3*x^8/8 +...+ A002203(n)^3*x^n/n +...
%e A212442 Also, the g.f. equals the infinite product:
%e A212442 A(x) = 1/( (1-2*x-x^2)^4 * (1-6*x^2+x^4)^16 * (1-14*x^3-x^6)^64 * (1-34*x^4+x^8)^280 * (1-82*x^5-x^10)^1344 * (1-198*x^6+x^12)^6496 *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) *...).
%e A212442 The exponents in these products begin:
%e A212442 A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].
%e A212442 The companion Pell numbers begin (at offset 1):
%e A212442 A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].
%o A212442 (PARI) /* Subroutine for the PARI programs that follow: */
%o A212442 {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o A212442 (PARI) /* G.F. by Definition: */
%o A212442 {a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}
%o A212442 (PARI) /* G.F. as a Finite Product: */
%o A212442 {a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*A002203(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}
%o A212442 (PARI) /* G.F. as an Infinite Product: */
%o A212442 {A212443(n)=(1/n)*sumdiv(n,d, moebius(n/d)*A002203(d)^2)}
%o A212442 {a(n)=polcoeff(1/prod(m=1,n, (1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^A212443(m)),n)}
%o A212442 for(n=0,30,print1(a(n),", "))
%Y A212442 Cf. A212443, A203803, A002203, A204062.
%K A212442 nonn,new
%O A212442 0,2
%A A212442 _Paul D. Hanna_, May 17 2012

%I A212443
%S A212443 4,16,64,280,1344,6496,32640,166320,862400,4523232,23970240,128063040,
%T A212443 689008320,3728973120,20285199872,110841302880,608029121280,
%U A212443 3346972244000,18480871268160,102328688556864,568014587806720,3160148362953120,17617881702072960
%N A212443 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.
%F A212443 G.f.: 1/Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} A002203(n)^3 * x^n/n), which equals the g.f. of A212442.
%o A212443 (PARI) {A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
%o A212443 {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*A002203(d)^2)/n)}
%o A212443 for(n=1,30,print1(a(n),","))
%Y A212443 Cf. A212442, A203853, A002203.
%K A212443 nonn,new
%O A212443 1,1
%A A212443 _Paul D. Hanna_, May 17 2012

%I A182168
%S A182168 3,8,2,6,8,3,4,3,2,3,6,5,0,8,9,7,7,1,7,2,8,4,5,9,9,8,4,0,3,0,3,9,8,8,
%T A182168 6,6,7,6,1,3,4,4,5,6,2,4,8,5,6,2,7,0,4,1,4,3,3,8,0,0,6,3,5,6,2,7,5,4,
%U A182168 6,0,3,3,9,6,0,0,8,9,6,9,2,2,3,7,0,1,3,7,8,5,3,4,2,2,8,3,5,4,7,1,4,8,4,2,4
%N A182168 Decimal expansion of imaginary part of i^(1/4).
%C A182168 Also sin(Pi/8) or sine of 22.5 degrees.
%C A182168 The real part of i^(1/4) or cos(Pi/8) is A144981.
%e A182168 0.382683432365089771728459984...
%Y A182168 Cf. A144981.
%K A182168 nonn,cons,new
%O A182168 0,1
%A A182168 _Stanislav Sykora_, May 16 2012

%I A212438
%S A212438 1,0,1,1,0,1,2,2,2,0,0,2,8,11,8,5,0,0,2,11,42,74,76,38,14,0,0,0,8,74,
%T A212438 296,633,768,558,219,50,0,0,0,5,76,633,2635,6134,8822,7916,4442,1404,
%U A212438 233
%N A212438 Irregular triangle read by rows: T(n,k) (n >= 4, k=4..2n-4) = number of polyhedra with n faces and k vertices.
%e A212438 Triangle begins:
%e A212438 1
%e A212438 0 1 1
%e A212438 0 1 2 2 2
%e A212438 0 0 2 8 11 8 5
%e A212438 0 0 2 11 42 74 76 38 14
%e A212438 0 0 0 8 74 296 633 768 558 219 50
%e A212438 0 0 0 5 76 633 2635 6134 8822 7916 4442 1404 233
%e A212438 ...
%Y A212438 A049337, A058787, A212438 are all versions of the same triangle.
%Y A212438 Row sums are A000944.
%Y A212438 Main diagonal is A002856.
%K A212438 nonn,tabf,new
%O A212438 4,7
%A A212438 _N. J. A. Sloane_, May 16 2012

%I A212436
%S A212436 9,3,3,0,9,2,0,7,5,5,9,8,2,0,8,5,6,3,5,4,0,4,1,0,1,7,1,4,0,8,7,4,3,5,
%T A212436 8,9,0,2,5,8,9,4,7,9,7,9,5,0,1,3,7,6,4,4,6,2,3,8,4,3,7,8,8,4,0,7,9,0,
%U A212436 6,7,2,1,6,6,3,3,0,1,2,4,3,4,3,0,1,7,6,7,3,6,3,0,3,2,7,4,3,3,6,3,7,4,8,7,6
%N A212436 Decimal expansion of the real part of e^(i/e).
%C A212436 Also cos(1/e).
%C A212436 The imaginary part of e^(i/e), or sin(1/e), is A212437.
%e A212436 0.933092075598208563540410171408743589...
%Y A212436 Cf. A212437, A085659, A085660.
%K A212436 nonn,cons,easy,new
%O A212436 0,1
%A A212436 _Stanislav Sykora_, May 16 2012

%I A212437
%S A212437 3,5,9,6,3,7,5,6,5,4,1,2,4,9,5,5,7,7,0,3,8,2,5,0,3,9,3,9,0,5,6,2,4,7,
%T A212437 1,3,7,3,7,9,5,8,1,9,5,1,7,7,9,0,5,8,1,6,6,6,3,9,7,4,7,1,1,4,2,0,3,0,
%U A212437 0,4,5,0,8,9,9,1,4,7,4,0,5,0,3,0,5,6,3,5,9,7,5,4,7,8,0,8,8,7,2,5,6,8,2,9,3
%N A212437 Decimal expansion of the imaginary part of e^(i/e).
%C A212437 Also sin(1/e).
%C A212437 The real part of e^(i/e), or cos(1/e) is in A212436.
%e A212437 0.3596375654124955770382503939056247...
%Y A212437 Cf. A212436, A085659, A085660.
%K A212437 nonn,cons,easy,new
%O A212437 0,1
%A A212437 _Stanislav Sykora_, May 16 2012

%I A201223
%S A201223 3,8,7,5,8,7,7,15,13,8,15,13,5,21,19,16,21,19,11,35,31,24,35,31,7,40,
%T A201223 37,33,40,37,13,48,43,35,48,43,16,55,49,39,55,49,9,65,61,56,65,61,32,
%U A201223 77,67,45,77,67,17,80,73,63,80,73,40,91,79,51,91,79,11,96,91
%N A201223 Primitive Eisenstein triples (a,b,c) listed as groups of three in order of increasing b.
%C A201223 An Eisenstein triple is a triple (a,b,c) of positive integers with a<c<b and a^2 - a*b + b^2 = c^2.  It is primitive if  a, b, and c are relatively prime.
%D A201223 Russell A. Gordon, Properties of Eisenstein Triples, Mathematics Magazine 85 (2012), 12-25.
%e A201223 (a,b,c)=(3,8,7) is an Eisenstein triple since 3<7<8 and 3^2 - 3*8 +8^2  = 7^2.  G.c.d. (3,8,7) = 1, so the triple is primitive.  No Eisenstein triple exists with b<8, so a(1)=3, a(2)=8, a(3)=7.
%t A201223 x = {}; For[b = 1, b <= 77, b++, For[c = 1, c < b, c++, For[a = 1, a < c, a++, {If[(a^2 - a*b + b^2 == c^2) && (GCD[a, b, c] == 1), AppendTo[x, {a, b, c}]]}]]]; Flatten[x]
%Y A201223 Cf. A121992.
%K A201223 nonn,new
%O A201223 1,1
%A A201223 _John W. Layman_, May 09 2012

%I A212267
%S A212267 1,1,2,1,4,16,1,6,72,272,1,8,168,2896,7936,1,10,304,10672,203904,
%T A212267 353792,1,12,480,26400,1198080,22112000,22368256,1,14,696,52880,
%U A212267 4071040,208521728,3412366336,1903757312,1,16,952,92912,10373760,976629760,51874413568
%N A212267 Array A(i,j) read by antidiagonals: A(i,j) is the (2*i-1)-th derivative of tan(tan(tan(...tan(x)))) nested j times evaluated at 0.
%C A212267 The determinant of the nXn such matrix has a closed form given in the Mathematica code below.
%C A212267 Rows appear to be given by polynomials (see formula section).
%H A212267 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NestedFunction.html">Nested Function</a>
%F A212267 A(i,j) = ((d/dx)^(2i-1) tan^j(x))_{x=0}.
%F A212267 Third row: n*(5*n - 1)*4 = 8*A005476(n).
%F A212267 Fourth row: 8/3*n*(11 - 84*n + 175*n^2).
%e A212267 Array A(i,j) begins:
%e A212267 .      1,        1,         1,         1,          1, ...
%e A212267 .      2,        4,         6,         8,         10, ...
%e A212267 .     16,       72,       168,       304,        480, ...
%e A212267 .    272,     2896,     10672,     26400,      52880, ...
%e A212267 .   7936,   203904,   1198080,   4071040,   10373760, ...
%e A212267 . 353792, 22112000, 208521728, 976629760, 3172514560, ...
%e A212267 Evaluate the (2*3-1)th derivate of tan(tan(tan(x))) at 0, which is 168. Thus A(3,3)=168.
%p A212267 A:= (i, j)-> (D@@(2*i-1))(tan@@j)(0):
%p A212267 seq (seq (A(i, 1+d-i), i=1..d), d=1..8); # _Alois P. Heinz_, May 13 2012
%t A212267 A[a_, b_] :=
%t A212267   A[a, b] =
%t A212267    Array[D[Nest[Tan, x, #2], {x, 2*#1 - 1}] /. x -> 0 &, {a, b}];
%t A212267 Print[A[7, 7] // MatrixForm];
%t A212267 Table2 = {};
%t A212267 k = 1;
%t A212267 While[k < 8, Table1 = {};
%t A212267   i = 1;
%t A212267   j = k;
%t A212267   While[0 < j,
%t A212267    AppendTo[Table1, First[Take[First[Take[A[7, 7], {i, i}]], {j, j}]]];
%t A212267    j = j - 1;
%t A212267    i = i + 1];
%t A212267   AppendTo[Table2, Table1];
%t A212267   k++];
%t A212267 Print[Flatten[Table2]];
%t A212267 Print[Table[Det[A[n, n]], {n, 1, 7}]];
%t A212267 Table[(2^(11/12 +
%t A212267        1/2 (5 + 3 (-1 + n)) (-1 + n)) 3^(-(1/2) (-1 +
%t A212267          n) n) Glaisher^3 \[Pi]^-n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]
%Y A212267 Columns j=1-3 give: A000182, A003718, A003720.
%K A212267 nonn,tabl,hard,new
%O A212267 1,3
%A A212267 _John M. Campbell_, May 12 2012
%E A212267 More terms from _Alois P. Heinz_, May 13 2012

%I A212261
%S A212261 1,1,1,1,2,1,1,3,12,1,1,4,33,128,1,1,5,64,731,1872,1,1,6,
%T A212261 105,2160,25857,37600,1,1,7,156,4765,121600,1311379,990784,1,1,
%U A212261 8,217,8896,368145,10138880,89060065,32333824,1
%V A212261 1,1,-1,1,-2,1,1,-3,12,-1,1,-4,33,-128,1,1,-5,64,-731,1872,-1,1,-6,
%W A212261 105,-2160,25857,-37600,1,1,-7,156,-4765,121600,-1311379,990784,-1,1,
%X A212261 -8,217,-8896,368145,-10138880,89060065,-32333824,1
%N A212261 Array A(i,j) read by antidiagonals: A(i,j) is the (2i-1)-th derivative of sin(sin(sin(...sin(x)))) nested j times evaluated at 0.
%C A212261 The determinant of the nXn such matrix has a closed form given in the formula section (and the Mathematica code below).
%C A212261 Rows appear to be given by polynomials (see formula section).
%F A212261 A(i,j) = ((d/dx)^(2i-1) sin^j(x))_{x=0}.
%F A212261 Let A_n denote the nXn such matrix. Then:
%F A212261 det(A_n)=(i^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2) G^3 (-(1/pi))^n B(1/2 + n) B(1 + n) B(3/2 + n))/e^(1/4), where B is the Barnes G-function and G is the Glaisher-Kinkelin constant (and i is the imaginary unit). (This can be shown by evaluating recurrence relations for det(A_n)). See Mathematica code below.
%F A212261 First row: 1.
%F A212261 Second row: -x.
%F A212261 Third row: x (5 x - 4).
%F A212261 Fourth row: -(1/3) x (164 + 7 x (-48 + 25 x)).
%F A212261 Fifth row: (8 - 7 x)^2 x (-24 + 25 x).
%F A212261 Sixth row: -(1/3) x (213568 - 766656 x + 1004696 x^2 - 572880 x^3 + 121275 x^4).
%F A212261 Seventh row: 1/3 x (-14371328 + 65012064 x - 111160192 x^2 + 91291200 x^3 - 36552516 x^4 + 5780775 x^5).
%F A212261 Second column: A003712.
%F A212261 Third column: A003715.
%e A212261 Evaluate the fifth derivative of sin(sin(sin(x))) at 0, which is 33. So the (3,3) entry of the array is 33. The array begins as:
%e A212261 |1   1       1         1          1          1         |
%e A212261 |-1  -2      -3        -4         -5         -6        |
%e A212261 |1   12      33        64         105        156       |
%e A212261 |-1  -128    -731      -2160      -4765      -8896     |
%e A212261 |1   1872    25857     121600     368145     873936    |
%e A212261 |-1  -37600  -1311379  -10138880  -42807605  -130426016|
%p A212261 A:= (i, j)-> (D@@(2*i-1))(sin@@j)(0):
%p A212261 seq (seq (A(i, 1+d-i), i=1..d), d=1..9); # _Alois P. Heinz_, May 14 2012
%t A212261 A[a_, b_] :=
%t A212261   A[a, b] =
%t A212261    Array[D[Nest[Sin, x, #2], {x, 2*#1 - 1}] /. x -> 0 &, {a, b}];
%t A212261 Print[A[7, 7] // MatrixForm];
%t A212261 Table2 = {};
%t A212261 k = 1;
%t A212261 While[k < 8, Table1 = {};
%t A212261   i = 1;
%t A212261   j = k;
%t A212261   While[0 < j,
%t A212261    AppendTo[Table1, First[Take[First[Take[A[7, 7], {i, i}]], {j, j}]]];
%t A212261    j = j - 1;
%t A212261    i = i + 1];
%t A212261   AppendTo[Table2, Table1];
%t A212261   k++];
%t A212261 Print[Flatten[Table2]]
%t A212261 Print[Table[Det[A[n, n]], {n, 1, 7}]];
%t A212261 Print[Table[(
%t A212261   I^(n + n^2) 2^(-(1/12) + n^2) 3^(n/2 - n^2/2)
%t A212261     Glaisher^3 (-(1/\[Pi]))^
%t A212261    n BarnesG[1/2 + n] BarnesG[1 + n] BarnesG[3/2 + n])/E^(1/4), {n, 1, 7}]]
%Y A212261 Cf. A003712, A003715.
%K A212261 sign,tabl,hard,nice,new
%O A212261 1,5
%A A212261 _John M. Campbell_, May 12 2012

%I A211133
%S A211133 1155,4599,5643,8415,9405,10395,10725,66495,200385,1100385,1769229,
%T A211133 2518725,2673585,2807805,4150575,4362115,8191161,8954495,11534489,
%U A211133 12645009,13435565,19875933,25560909,31122325,45285075,45930885,55257111,62379225,75081435,93205889
%N A211133 Odd sides belonging to multiple primitive Euler bricks.
%C A211133 Primitive Euler bricks are defined by (a, b, c) where a^2+b^2=u^2, b^2+c^2=v^2, a^2+c^2=w^2, gcd(a, b, c) = 1, and all values are integers, thus defining a cuboid with integer edges and face diagonals.
%C A211133 At most one of a, b, and c must be odd. This list shows those odd values which belong to more than one valid (a, b, c) set.
%H A211133 Eric Weisstein, <a href="http://mathworld.wolfram.com/EulerBrick.html">MathWorld: Euler Brick</a>
%K A211133 nonn,new
%O A211133 1,1
%A A211133 _Alan Griffiths_, May 11 2012

%I A212417
%S A212417 1,1,1,3,7,35,135,945,5193,46737
%N A212417 Size of the equivalence classes of S_n containing the identity permutation under transformations of positionally adjacent elements of the form abc <--> acb <--> bac where a<b<c.
%D A212417 J. Cassaigne, M. Espie, D. Krob, J.-C. Novelli, F. Hivert: The Chinese Monoid, Int'l. J. Algebra and Comp. 11 (2001), 301-334.
%H A212417 S. Linton, J. Propp, T. Roby, and J. West: Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, 2011. <a href="http://arxiv.org/abs/1111.3920">arXiv:1111.3920 [math.CO]</a>.
%e A212417 Contribution from _Alois P. Heinz_, May 16 2012: (Start)
%e A212417 a(3) = 3: {123, 132, 213}.
%e A212417 a(4) = 7: {1234, 1243, 1324, 1423, 2134, 2143, 2314}. (End)
%Y A212417 Cf. A210667, A210668, A210669, A210671, A212417.
%K A212417 nonn,more,new
%O A212417 0,4
%A A212417 _Tom Roby_, May 15 2012
%E A212417 a(0)-a(2), a(9) from _Alois P. Heinz_, May 16 2012

%I A182405
%S A182405 0,8,10,24,28,34,46,52,58,66,78,80,94,96,126,134,162,166,180,208,240,
%T A182405 258,270
%N A182405 Records of A164368(n) - A194598(n).
%C A182405 Theorem. If in the intervals {(A194598(n), A164368(n))} with lengths a(n)-1 the number of primes is unbounded, then there exist arbitrary long sequences of consecutive primes p_k, p_(k+1),...,p_m such that every interval (p_i/2, p_(i+1)/2), i=k,k+1,...,m-1, contains a prime.
%Y A182405 Cf. A164368, A194598, A182366.
%K A182405 nonn,new
%O A182405 1,2
%A A182405 _Vladimir Shevelev_, Apr 27 2012

%I A212426
%S A212426 1,1,6,58,704,9765,147870,2382666,40211406,703391832,12660466838,
%T A212426 233250098568,4381343112174,83656137696686,1619826435181890,
%U A212426 31746893434751700,628829271360548586,12572386589515935168,253455923387917072890
%N A212426  a(n) = A212425(n) / (2*n-1).
%F A212426  Given g.f. A(x), then G(x) = d/dx A(x^8)/(4*x^4) = Sum_{n>=1} (2*n-1)*a(n)*x^(8*n-5) is the g.f. of A212425 and satisfies: G(x) = (x + G(G(x)))^3.
%e A212426  G.f.: A(x) = x + x^2 + 6*x^3 + 58*x^4 + 704*x^5 + 9765*x^6 + 147870*x^7 +...
%e A212426 Let G(x) = d/dx A(x^8)/(4*x^4), then G(x) = (x + G(G(x)))^3, where
%e A212426 G(x) = x^3 + 3*x^11 + 30*x^19 + 406*x^27 + 6336*x^35 + 107415*x^43 +...
%e A212426 G(G(x)) = x^9 + 9*x^17 + 117*x^25 + 1788*x^33 + 29925*x^41 + 530910*x^49 +...
%o A212426  (PARI) {a(n)=local(G=x^3+3*x^11); for(i=1, n, G=(x+subst(G, x, G +O(x^(8*n))))^3); polcoeff(G, 8*n-5)/(2*n-1)}
%o A212426 for(n=1, 30, print1(a(n), ", "))
%Y A212426  Cf. A212425, A212391.
%K A212426 nonn,new
%O A212426 1,3
%A A212426 _Paul D. Hanna_, May 16 2012

%I A212425
%S A212425 1,3,30,406,6336,107415,1922310,35739990,683593902,13364444808,
%T A212425 265869803598,5364752267064,109533577804350,2258715717810522,
%U A212425 46974966620274810,984153696477302700,20751365954898103338,440033530633057730880,9377869165352931696930
%N A212425 G.f. satisfies: A(x) = ( x + A(A(x)) )^3 where g.f. A(x) = Sum_{n>=1} a(n)*x^(8*n-5).
%C A212425 Conjecture: (2*n-1) divides a(n); see A212426.
%C A212425 More generally, we have the conjecture:
%C A212425 If    A(x) = ( x + A(A(x)) )^b
%C A212425 where A(x) = Sum_{n>=1} a(n) * x^((b^2-1)*(n-1)+b)
%C A212425 then  ((b-1)*(n-1)+1) divides a(n).
%H A212425 Alois P. Heinz, <a href="/A212425/b212425.txt">Table of n, a(n) for n = 1..100</a>
%F A212425 G.f.: A(x) = d/dx G(x^8)/(4*x^4) where G(x) = Sum_{n>=1} A212426(n)*x^n is the g.f. of A212426.
%F A212425 a(n) = (2*n-1)*A212426(n).
%F A212425 a(n) = T(8*n-5,1), T(n,k) = if n<3*k then 0 else if n/3=k then 1 else sum(j=0..3*k-1, C(3*k,j)*sum(i=3*k-j+1..n-j-1, T(i,3*k-j)*T(n-j,i))). [_Vladimir Kruchinin_, May 17 2012]
%e A212425 G.f.: A(x) = x^3 + 3*x^11 + 30*x^19 + 406*x^27 + 6336*x^35 + 107415*x^43 +...
%e A212425 such that A(x) = (x + A(A(x)))^3, where
%e A212425 A(A(x)) = x^9 + 9*x^17 + 117*x^25 + 1788*x^33 + 29925*x^41 + 530910*x^49 + 9809193*x^57 + 186734493*x^65 + 3637247445*x^73 +...
%e A212425 Note that A(A(x))^(1/3) = A(x) + A(A(A(x))), where
%e A212425 A(A(x))^(1/3) = x^3 + 3*x^11 + 30*x^19 + 407*x^27 + 6363*x^35 + 108009*x^43 + 1934721*x^51 + 35995815*x^59 + 688861845*x^67 +...
%e A212425 A(A(A(x))) = x^27 + 27*x^35 + 594*x^43 + 12411*x^51 + 255825*x^59 + 5267943*x^67 + 108864873*x^75 + 2261456685*x^83 +...
%p A212425 A:= proc(n) option remember;
%p A212425       `if`(n=1, unapply(x, x), unapply (convert (series
%p A212425        ((x+(A(n-1)@@2)(x))^3, x, n+10), polynom), x))
%p A212425     end:
%p A212425 a:= n-> coeff (A(8*n-5)(x), x, 8*n-5):
%p A212425 seq (a(n), n=1..30);  # _Alois P. Heinz_, May 17 2012
%o A212425 (PARI) {a(n)=local(A=x^3+3*x^11); for(i=1, n, A=(x+subst(A, x, A+O(x^(8*n))))^3); polcoeff(A, 8*n-5)}
%o A212425 for(n=1, 30, print1(a(n), ", "))
%o A212425 (Maxima) T(n,k):= if n<3*k then 0 else if n/3=k then 1 else sum(binomial(3*k,j)*sum(T(i,3*k-j)*T(n-j,i), i,3*k-j+1,n-j-1), j,0,3*k-1);
%o A212425 makelist(T(n,1),n,1,20); [_Vladimir Kruchinin_, May 17 2012]
%Y A212425 Cf. A212426, A212392.
%K A212425 nonn,new
%O A212425 1,2
%A A212425 _Paul D. Hanna_, May 16 2012

%I A212328
%S A212328 1,1,4,4,58,139,139,1597,1597,8158,8158,67207,67207,598648,2192971,
%T A212328 6975940,21324847,21324847,21324847,408745336,1571006803,8544575605,
%U A212328 29465282011,29465282011,217751639665,500181176146,1347469785589,6431201442247,6431201442247
%N A212328 Smallest k such that k^3 + 17 is divisible by 3^n.
%C A212328 This sequence is generalizable :  the smallest k such that k^3 + p is divisible by 3^n exists if the prime p is congruent to + - 1 mod 18. For example, the sequence with p = 19 is given by {2, 2, 2, 20, 20, 20, 263, 992, 3179, 16301, 55667, 173765, 528059, …}. (See A129805). This sequence is given with the smallest p = 17.
%H A212328 Charles R Greathouse IV, <a href="/A212328/b212328.txt">Table of n, a(n) for n = 1..1000</a>
%e A212328 a(4) = 4 because 4^3 + 17 = 81 is divisible by 3^4.
%p A212328 with(numtheory):for n from 1 to 20 do:i:=0:for x from 1 to 10^8 while(i=0) do: z:= x^3 + 17:if irem(z,3^n)=0 then i:=1: printf ( "%d %d \n",n,x):else fi:od:od:
%o A212328 (PARI) print1(k=1);for(n=2,100,if(Mod(k,3^n)^3!=-17,k+=3^(n-2)* if(Mod(k+3^(n-2),3^n)^3==-17,1,2));print1(", "k)) \\ _Charles R Greathouse IV_, May 14 2012
%Y A212328 Cf. A129805.
%K A212328 nonn,easy,new
%O A212328 1,3
%A A212328 _Michel Lagneau_, May 14 2012
%E A212328 a(20)-a(29) from _Charles R Greathouse IV_, May 14 2012

%I A212325
%S A212325 167,163,157,149,139,127,113,97,79,59,37,13,13,41,71,103,
%T A212325 137,173,211,251,293,337,383,431,481,533,587,643,701,761,823,887,953,
%U A212325 1021,1091,1163,1237,1313,1391,1471,1553,1637,1723,1811,1901,1993,2087,2183
%V A212325 -167,-163,-157,-149,-139,-127,-113,-97,-79,-59,-37,-13,13,41,71,103,
%W A212325 137,173,211,251,293,337,383,431,481,533,587,643,701,761,823,887,953,
%X A212325 1021,1091,1163,1237,1313,1391,1471,1553,1637,1723,1811,1901,1993,2087,2183
%N A212325 Prime-generating polynomial: n^2 + 3*n - 167.
%C A212325 The polynomial generates 24 primes in absolute value (23 distinct ones) in row starting from n=0 (and 42 primes in absolute value for n from 0 to 46).
%C A212325 The polynomial n^2 - 49*n + 431 generates the same primes in reverse order.
%C A212325 Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 677) the polynomial 13*n^2 - 311*n + 1847 (13*n^2 - 469*n + 4217) generating 23 primes and two noncomposite numbers (in absolute value) in row starting from n=0 (1847, 1549, 1277, 1031, 811, 617, 449, 307, 191, 101, 37, -1, -13, 1, 41, 107, 199, 317, 461, 631, 827, 1049, 1297, 1571, 1871).
%C A212325 Note: another interesting algorithm to produce prime-generating polynomials could be N = m*n^2 + (6*m+1)*n + 8*m + 3, where m, 6*m+1 and 8*m+3 are primes. For m=7 then n=t-20 we get N = 7*t^2 - 237*t + 1999, which generates the following primes: 239, 163, 101, 53, 19, -1, -7, 1, 23, 59, 109, 173, 251 (we can see the same pattern: …, -1, -m, 1, …).
%H A212325 Bruno Berselli, <a href="/A212325/b212325.txt">Table of n, a(n) for n = 0..1000</a>
%H A212325 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial">MathWorld: Prime-Generating Polynomial</a>
%H A212325 <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A212325 G.f.: (-167+338*x-169x^2)/(1-x)^3. [_Bruno Berselli_, May 18 2012]
%o A212325 (MAGMA) [n^2+3*n-167: n in [0..47]]; // _Bruno Berselli_, May 18 2012
%K A212325 sign,easy,new
%O A212325 0,1
%A A212325 _Marius Coman_, May 14 2012
%E A212325 Edited from _Bruno Berselli_, May 18 2012

%I A212324
%S A212324 362,37692,55185580,758665388,641947636,8948910312,126203947828,
%T A212324 1374427747336,19738612219080,285537263797392,255480709713456,
%U A212324 4156846359584754,3719283584891622,54448930157994828,801081711581734764,661763153045780892,9779554604050169400
%N A212324 (1/p)*Sum_{k=0..n}binomial(n,k)^4 where p is a prime in the interval ]n, 4n/3]
%C A212324 For any p in the interval ]n, 4n/3], p divides Sum_{k=0..n}binomial(n,k)^4 (see the link).
%H A212324 Peter Vandendriessche and Hojoo Lee, <a href="http://www.scribd.com/doc/24487088/Hojoo-Lee-Peter-Vandendriessche-Problems-in-Elementary-Number-Theory">Problems in elementary number theory</a>, Problem E16
%e A212324 362 is in the sequence because, for n=4, 5 is in the interval ]4, 16/3] and (1/5)*Sum_{k=0..4}binomial(4,k)^4 =(1/5)*(1^4 + 4^4 + 6^4 + 4^4 + 1^4) = 1810/5 = 362.
%p A212324 with(numtheory): for n from 1 to 20 do: for p from n+1 to floor(4*n/3) do: if type(p,prime)=true then s:=sum(binomial(n,k)^4,k=0..n):s:=s/p: printf(`%d, `,s):else fi:od:od:
%K A212324 nonn,new
%O A212324 1,1
%A A212324 _Michel Lagneau_, May 14 2012

%I A212276
%S A212276 2,17,37,15101701
%N A212276 Prime generalized Catalan numbers.
%C A212276 A004148(n) for n = 3, 6, 7, 22, no more through 70. Note that Koshy and Salmassi give an elementary proof that the only prime Catalan numbers A000108 are a(2) = 2 and a(3) = 5.
%C A212276 The next term, if it exists, has more than 2000 digits. - _Charles R Greathouse IV_, May 13 2012
%F A212276 A004148 INTERSECTION A000040.
%Y A212276 Cf. A000040, A000108, A004148, A092840.
%K A212276 nonn,new
%O A212276 1,1
%A A212276 _Jonathan Vos Post_, May 13 2012

%I A212266
%S A212266 59,73,79,89,101,109,197,211,239,241,263,281,307,337,367,373,379,409,
%T A212266 419,421,439,443,449,461,463,491,523,547,557,571,593,601,613,617,631,
%U A212266 647,653,659,673,701,709,769,797,811,839,853,863,881,907,929,937,941,967
%N A212266 Primes p such that p - m! is composite, where m is the greatest number such that m! < p.
%C A212266 The first five terms 59,73,79,89,101 belong to A023209.
%C A212266 The terms 409,419,421,439,443,449 also belong to A127209
%H A212266 Charles R Greathouse IV, <a href="/A212266/b212266.txt">Table of n, a(n) for n = 1..10000</a>
%e A212266 29 is not a member because 29-4! = 5 is prime, not composite.
%e A212266 59 is a member because 59-4! = 35 is composite.
%o A212266 (PARI) for(n=3,5,N=n!;forprime(p=N+3,N*(n+1),if(!isprime(p-N), print1(p", ")))) \\ _Charles R Greathouse IV_, May 12 2012
%o A212266 (PARI) is_A212266(p)=isprime(p) & for(n=1,p, n!<p || return(bigomega(p-(n-1)!)>1))  \\ _M. F. Hasler_, May 20 2012
%Y A212266 Cf. A212600, A212598, A136437, A023209, A127209.
%K A212266 nonn,easy,new
%O A212266 1,1
%A A212266 _Balarka Sen_, May 12 2012

%I A212269
%S A212269 2,5,19,205,3011,92875,4763459,459630701,78223965193,24270274906085,
%T A212269 13497818986883771,13571363009654254429,24562890586806439035377,
%U A212269 80199120146273882569630015,471874707649862024071657639861,5005895207027974222377733802848093
%N A212269 Number of ways to place k non-attacking kings on an n x n cylindrical chessboard, summed over all k >= 0.
%H A212269 Vaclav Kotesovec, <a href="/A212269/b212269.txt">Table of n, a(n) for n = 1..23</a>
%H A212269 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Non-attacking chess pieces</a>
%F A212269 Limit n ->infinity (a(n))^(1/n^2) = 1.342643951124...
%Y A212269 Cf. A063443(n+1), A067958, A194650-A194654, A182407.
%K A212269 nonn,hard,new
%O A212269 1,1
%A A212269 _Vaclav Kotesovec_, May 12 2012

%I A212270
%S A212270 2,7,43,933,36211,3557711,746156517,363549830913,394677987525997,
%T A212270 974602314570939359,5418730454986467701985,68176187476467835406646029,
%U A212270 1936241516342334422813929891295,124281423643836238320564876791634465,18018270577720149773239661332878801006033
%N A212270 Number of ways to place k non-attacking wazirs on an n x n cylindrical chessboard, summed over all k >= 0.
%C A212270 Wazir is a leaper [0,1].
%H A212270 Vaclav Kotesovec, <a href="/A212270/b212270.txt">Table of n, a(n) for n = 1..24</a>
%H A212270 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Non-attacking chess pieces</a>
%F A212270 Limit n ->infinity (a(n))^(1/n^2) is the hard square entropy constant A085850.
%Y A212270 Cf. A006506, A027683(n-1), A182407, A212269, A212271.
%K A212270 nonn,hard,new
%O A212270 1,1
%A A212270 _Vaclav Kotesovec_, May 12 2012

%I A212271
%S A212271 2,9,80,1600,79033,8156736,2055960192,1108756350625,1411080429618656,
%T A212271 3943472747846953216,25425527581172360096017,
%U A212271 365481944233773616212640000,11980566143208960475692367828480,882106482533191605447029340350009049,147314997388032765439791110273770608260928
%N A212271 Number of ways to place k non-attacking ferses on an n x n cylindrical chessboard, summed over all k >= 0.
%C A212271 Fers is a leaper [1,1].
%H A212271 Vaclav Kotesovec, <a href="/A212271/b212271.txt">Table of n, a(n) for n = 1..17</a>
%H A212271 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Non-attacking chess pieces</a>
%F A212271 Limit n ->infinity (a(n))^(1/n^2) is the hard square entropy constant A085850.
%Y A212271 Cf. A067965, A067960, A182407, A212269, A212270.
%K A212271 nonn,hard,new
%O A212271 1,1
%A A212271 _Vaclav Kotesovec_, May 12 2012

%I A182521
%S A182521 2,3,5,7,10,17,23,25,45,47,77,87,95,103,107,137,143,175,215,247,283,
%T A182521 287,313,347,355,373,385,397,425,443,455,467,565,577,593,637,653,667,
%U A182521 703,737,773,775,787,850,907,913,917,943,975
%N A182521 Numbers n such that A182481(n)=1 and there is not a representation n=d_1*d_2 with d_2>1, such that A182481(d_1)=d_2.
%C A182521 Or the numbers n such that 6*n-1 is lesser of twin primes which occurs in A182482 only once.
%C A182521 All terms of A060212 are in the sequence.
%F A182521 Numbers n for which A182483(A182513(n))/n = A182481(n) = 1.
%Y A182521 Cf. A182481, A182482, A182483, A182513, A060212.
%K A182521 nonn,new
%O A182521 1,1
%A A182521 _Vladimir Shevelev_, May 03 2012

%I A212372
%S A212372 1,4,6,8,9,10,20,21,30,32,40,42,50,51,52,54,60,62,63,64,65,70,72,74,
%T A212372 75,76,80,81,82,84,85,86,87,90,91,92,93,94,95,96,98,210,310,320,321,
%U A212372 410,420,430,432,510,520,530,531,532,540,542,543,610,620,621,630
%N A212372 Nonprime numbers with distinct digits in descending order.
%C A212372 Sequence is finite with 935 terms, last term is a(935) = 9876543210.
%C A212372 Complement of A052014 with respect to A009995.
%H A212372 Jaroslav Krizek, <a href="/A212372/b212372.txt">Table of n, a(n) for n = 1..935</a>(complete list).
%F A212372  A178788(a(n)) = 1.
%Y A212372 Cf. A052014 (primes with distinct digits in descending order), A009995 (numbers with distinct digits in descending order).
%K A212372 nonn,base,fini,new
%O A212372 1,2
%A A212372 _Jaroslav Krizek_, May 10 2012

%I A212322
%S A212322 1,1,1,3,3,5,13,17,29,55,99,161,293,507,881,1561,2727,4743,8337,14579,
%T A212322 25497,44675,78173,136753,239437,419077,733377,1283701,2246823,
%U A212322 3932249,6882603,12046313,21083545,36901587,64586887,113042011,197851265,346287829,606086169
%N A212322 Number of compositions of n so that no two adjacent parts are equal, and the first part is not equal to the last part if there is more than one part.
%C A212322 Also known as cyclic Carlitz compositions.
%D A212322 Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010, pages 87-88.
%H A212322 Jair Taylor, <a href="/A212322/b212322.txt">Table of n, a(n) for n = 0..199</a>
%F A212322 G.f.:  1 + sum(k>0: x^k/(1+x^k)^2)/(1 - sum(k>0, x^k/(1+x^k))) + sum(k>0, x^(2k)/(1 + x^k))
%e A212322 The cyclic Carlitz compositions of the n = 1...6 are
%e A212322 1;
%e A212322 2;
%e A212322 12, 21, 3;
%e A212322 13, 31, 4;
%e A212322 14, 23, 32, 41,5;
%e A212322 1212, 123, 132, 15, 2121, 213, 231, 24, 312, 321, 42, 51, 6.
%o A212322 (Sage)
%o A212322 for n in range(15):
%o A212322 ... Q = []
%o A212322 ... for comp in Compositions(n) :
%o A212322 ...... if len(comp) == 1 or all([ comp[k] != comp[k+1] for k in range(-1,len(comp)-1) ]):
%o A212322 ......... Q.append(comp)
%o A212322 ... print len(Q), ",",
%Y A212322 Removing restriction on the first and last parts gives the Carlitz compositions, A003242.
%K A212322 nonn,new
%O A212322 0,4
%A A212322 _Jair Taylor_, May 13 2012

%I A182585
%S A182585 34,169,194,985,4181,9077,14701,51641,135137,294685,499393,646018,
%T A182585 925765,1136689,1346269,2012674,6625109,8399329,9647009,11485154,
%U A182585 16964653,21531778,43484701,48928105,111242465,137295677,144059117,165580141,225058681,253191266
%N A182585 Markov numbers that are semiprime.
%C A182585 This is to A178444 as semiprimes A001358 are to primes A000040.
%F A182585 A001358 Intersection A002559. {x, y, or z satisfying x^2 + y^2 + z^2 = 3xyz, also such that A001222(of that number) = 2}.
%e A182585 194 is in this sequence because it is a Markoff (or Markov) number, and 194 = 2 * 97.
%Y A182585 Cf. A001358, A002559, A178444.
%K A182585 nonn,easy,new
%O A182585 1,1
%A A182585 _Jonathan Vos Post_, May 06 2012

%I A210767
%S A210767 11,12,14,16,21,23,25,29,32,34,38,41,43,47,52,58,61,67,74,76,83,85,89,
%T A210767 92,98,101,102,104,106,110,111,113,119,120,131,133,140,146,160,164,
%U A210767 166,179,191,197,201,203,205,209,210,223,230,232,250,269,290,296,302
%N A210767 Numbers whose digit sum as well as sum of the 4th powers of the digits is a prime.
%C A210767 This is to the exponent 4 as A182404 is to the exponent 2.
%H A210767 Charles R Greathouse IV, <a href="/A210767/b210767.txt">Table of n, a(n) for n = 1..10000</a>
%F A210767 {n such that A055013(n) and A007953(n) are both primes}.
%e A210767 21 is in the sequence because sum of digits 2+1= 3 is prime, and sum of the 4th powers of the digits 2^4+1^4=17 is a prime.
%o A210767 (PARI) dspow(n,b,k)=my(s);while(n,s+=(n%b)^k;n\=b);s
%o A210767 select(n->isprime(sumdigits(n))&&isprime(dspow(n,10,4)), vector(10^3, i, i)) \\ _Charles R Greathouse IV_, May 11 2012
%Y A210767 Cf. A007953, A055013, A182404.
%K A210767 nonn,base,easy,new
%O A210767 1,1
%A A210767 _Jonathan Vos Post_, May 10 2012

%I A212354
%S A212354 3,10,10,20,21,36,41,55,59,61,59,55,92,105,118,96,92,126,171,152,105,
%T A212354 175,188,152,136,175,168,254,215,300,215,242,242,197,238,331,365,210,
%U A212354 337,406,343,415,402,254,358,403,296,337,327,300,554,538,595,405
%N A212354 a(n) is the second smallest positive incongruent solutions of the congruence x^2 + (x+1)^2 == 0 (mod prime), where prime = A002144(n).
%C A212354 The companion sequence is A212353.
%C A212354 See the comments on A212353 for the proof of two incongruent solutions of this congruence for each prime A002144(n). One takes the smallest positive representatives in each case as A21235(n) and a(n), with A21235(n) < a(n).
%C A212354 All positive solutions of this congruence are provided by the two sequences with entries u(n) = A212353(n) (mod A002144(n)) and v(n) = a(n) (mod A002144(n)). For the cases p = 5, 13 and 17 see A047219, A212160 and A212161, respectively, where the even indexed numbers are the u(n) and the odd indexed ones the v(n) (bisection).
%C A212354 r2(n) := 2*a(n) + 1 >= A002144(n) iff r(n) := 2*A212353(n) + 1 <= A002144(n)- 2. r2(n)^2 == +1 (Modd A002144(n)) but only r(n) belongs to the relevant restricted residue class.  See A206549. Note that floor(r2(n)^2/A002144(n)) is odd. The same holds for r2 replaced by r.
%F A212354 a(n) is the second smallest positive incongruent solution of the congruence x^2 + (x+1)^2 = 2*x^2 + 2*x +1  == 0 (mod A002144(n)), where A002144 lists the primes 1 modulo 4.
%F A212354 a(n) = A002144(n) - 1 - A212353(n), n>=1.
%e A212354 n=1: a(1)=3 because 3^2+4^2 = 25 == 0 (mod 5). The other solution is (5-1)-3= 1 = A212353(1).
%e A212354 n=3: a(3)=10 because 10^2+11^2 = 221 = 13*17 == 0 (mod 17). (17-1)-10 = 6 =  A212353(3).
%e A212354 n=14: a(14)=105 because p=A002144(14) = 113 = A027862(5), and  105^2+106^2 =  197*113 == 0 (mod 113). (113-1)-105 = 7 = A212353(14).
%e A212354 The first pair of solutions [u(n)=A212353(n), v(n)=a(n)], n>=1, are:
%e A212354   [1, 3], [2, 10], [6, 10], [8, 20], [15, 21], [4, 36],
%e A212354   [11, 41], [5, 55], [13, 59], [27, 61],...
%Y A212354 Cf. 212353, A206549.
%K A212354 nonn,new
%O A212354 1,1
%A A212354 _Wolfdieter Lang_, May 10 2012

%I A212353
%S A212353 1,2,6,8,15,4,11,5,13,27,37,45,16,7,18,52,64,46,9,40,91,53,44,88,120,
%T A212353 93,108,26,77,12,101,94,106,155,134,57,31,190,71,14,89,33,54,206,150,
%U A212353 117,244,219,241,276,38,62,17,211,243,74,277,307,325,67,306,176,43
%N A212353 a(n) is the smallest positive solutions of the congruence x^2 + (x+1)^2 == 0 (mod prime), where prime = A002144(n).
%C A212353 The companion sequence is A212354.
%C A212353 There are at most two incongruent solutions of this congruence due to the degree. The fact that there are precisely two such solutions for each prime of the form 4*k+1 (see A002144) is due to the reduction of this problem to one of quadratic residues, namely to X^2 == -1 mod(2p), with p a prime (see the Nagell reference, given in A210848, pp. 132-3, especially theorem 77,  adapted to the quadratic form f(x) = 2*x^2 + 2*x +1, with discriminant D=-4. This congruence with composite modulus has exactly two incongruent solutions because X^2==-1 (mod 2) has only the solution +1 modulo 2 (odd numbers), and X^2 == -1  (mod p) has (at least one) solution if the Legendre symbol (-1/p) = +1 (i.e. if -1 is a quadratic residue modulo p). Now (-1/p) = (-1)^(p-1)/2 (see, e.g., the Niven-Zuckerman-Montgomery reference given in A001844, Theorem 3.2  (1), p. 132). Hence there is a solution modulo p iff p==1 (mod 4). Call the smallest positive one X0, with 0<  X0 < p-1 . Then one also has  the incongruent solution X1:= p-X0. This implies that there are precisely two incongruent solution of the original congruence  modulo 2*p for each 1 (mod 4) prime (see, e.g., Nagell's book. pp.83-4, Theorem 46). If u is a solution for p= A002144(n) (the existence of u has just been proven) then also the companion v:=p-1-u  satisfies this congruence, and v is incongruent to u modulo p.
%C A212353 Note that x^2 + (x+1)^2 = 4*T(x) +1, with the triangular numbers A000217.
%C A212353 The primes with x^2 +(x+1)^2 = prime (necessarily from  A002144) are found under A027862. The corresponding x values are found under A027861. These x values explain the positions n' where a(n') is smaller than a(n'-1) (for n'>=6): determine  k with  x=A027861(k), and then n' from A027862(k) = A002144(n'). Note that a(n') = x for such values n'. E.g., n'=6 with a(6)=4:  x=4=A027861(3), p=41=A027862(3) = A002144(6). These values n' are n' = 1, 2, 6, 8, 14, 19, 30,...
%C A212353 All positive solutions of this congruence are provided by the two sequences with entries u(n) = a(n) (mod A002144(n)) and v(n) = A212354(n) (mod A002144(n)), n>=1. For the cases p = 5, 13 and 17 see A047219, A212160 and A212161, respectively, where the even indexed numbers are the u(n) and the odd indexed ones the v(n) (bisection).
%C A212353 2*a(n) + 1 = A206549(n), the smallest positive nontrivial solution of X^2 == +1 (Modd A002144(n)). For the next larger solution 2*A212354(n) + 1 >= p, hence it does not belong to the restricted residue system Modd A002144(n).
%F A212353 a(n) is the smaller of the two smallest positive incongruent solutions of the congruence x^2 + (x+1)^2 = 2*x^2 + 2*x + 1  == 0 (mod A002144(n)), where A002144 lists the primes
%F A212353   1 modulo 4 (primes of the form 4*k+1). For the proof of  the existence of a(n) see a comment above. The next larger incongruent companion solution is A212354(n), n>=1.
%e A212353 n=1: a(1)=1 because 1^2+2^2 = 5 == 0 (mod 5). The companion solution is (5-1)-1= 3 = A212354(1).
%e A212353 n=3: a(3)=6 because 6^2+7^2 = 85 = 5*17 == 0 (mod 17). The companion is (17-1)-6 = 10 =  A212354(3).
%e A212353 n=14: a(14)=7 because p=A002144(14) = 113 = A027862(5), and  49^2+50^2 = 113. The companion is (113-1)-7 = 105 = A212354(14).
%Y A212353 Cf. A047219(1)=a(1), A212160(1)=a(2), A212161(1)=a(3), A212354 (companions), A206549.
%K A212353 nonn,new
%O A212353 1,2
%A A212353 _Wolfdieter Lang_, May 10 2012

%I A212223
%S A212223 2,6,6,882,1386,2007986541
%N A212223 First a(n) > 1 whose expansions in prime bases 2 to prime(n) have the same sum of digits.
%C A212223 Case a(1) is trivial since only base p(1)=2 is involved.
%C A212223 Conjecture: the sequence never terminates.
%e A212223 a(5) = 1386 because that number has the same sum of digits in the first 5 prime bases 2, 3, 5, 7, 11 (see A212222 and A000040).
%Y A212223 Cf. A212222, A135127, A135121, A037301.
%Y A212223 Cf. A000040.
%K A212223 nonn,base,new
%O A212223 1,1
%A A212223 _Stanislav Sykora_, May 10 2012

%I A210733
%S A210733 1,2,3,4,6,6,12,9,10,10,27,13,15,22,16,16,18,19,39,22,22,43,24,24,48,
%T A210733 26,28,43,32,38,75,33,36,58,35,57,55,47,48,52,54,72,52,51,72,54,72,53,
%U A210733 64,62,52,52,63,60,55,60,316,70,63,68,64,96,66,115,66,92
%N A210733 Least k > n-1 such that 4^n + 2^k - 1 is a prime number
%C A210733 as n increases k is in average 1.43*n.
%H A210733 Pierre CAMI, <a href="/A210733/b210733.txt">Table of n, a(n) for n = 1..3867</a>
%e A210733 4^1+2^1-1=5 prime so a(1)=1.
%e A210733 4^2+2^2-1=19 prime so a(2)=2.
%e A210733 4^3+2^3-1=71 prime so a(3)=3.
%t A210733 Table[k = n; While[! PrimeQ[4^n + 2^k - 1], k++];  k, {n, 100}] (* _T. D. Noe_, May 16 2012 *)
%o A210733 PFGW64 from primeform group and SCRIPTIFY
%o A210733 Command : PFGW64 -f in.txt
%o A210733 in.txt file :
%o A210733 SCRIPT
%o A210733 DIM nn,0
%o A210733 DIM kk
%o A210733 DIMS tt
%o A210733 OPENFILEOUT myfile,a(n).txt
%o A210733 LABEL loopn
%o A210733 SET nn,nn+1
%o A210733 SET kk,nn-1
%o A210733 LABEL loopk
%o A210733 SET kk,kk+1
%o A210733 SETS tt,%d,%d\,;nn;kk
%o A210733 PRP 4^nn+2^kk-1,tt
%o A210733 IF ISPRP THEN GOTO a
%o A210733 IF ISPRIME THEN GOTO a
%o A210733 GOTO loopk
%o A210733 LABEL a
%o A210733 WRITE myfile,tt
%o A210733 GOTO loopn
%K A210733 nonn,new
%O A210733 1,2
%A A210733 _Pierre CAMI_, May 10 2012

%I A210732
%S A210732 6,9,15,18,21,24,27,30,31,33,37,39,43,44,46,47,53,56,57,62,65,66,70,
%T A210732 73,74,75,76,78,81,83,86,88,90,91,92,93,97,99,102,103,106,107,109,110,
%U A210732 114,116,117,118,119,121,122,123,125,126,127,129,131,133,135,136
%N A210732 Numbers n for which sigma*(n)=sigma*(x)+sigma*(y), where n=x+y and sigma*(n) is the sum of the anti-divisors of n.
%C A210732 Similar to A211223 but using anti-divisors.
%e A210732 sigma*(127)=sigma*(45)+sigma*(82) that is 212=86+126.
%e A210732 In more than one way:
%e A210732 sigma*(133)=sigma*(50)+sigma*(83)=sigma*(52)+sigma*(81) that is
%e A210732 204=80+124=94+110.
%p A210732 with(numtheory);
%p A210732 A210732:=proc(q)
%p A210732 local a,b,c,i,j,k,n;
%p A210732 for n from 3 to q do
%p A210732   a:=0;
%p A210732   for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
%p A210732   for i from 1 to trunc(n/2) do
%p A210732    b:=0; c:=0;
%p A210732    for k from 2 to i-1 do if abs((i mod k)-k/2)<1 then b:=b+k; fi; od;
%p A210732    for k from 2 to n-i-1 do if abs(((n-i) mod k)-k/2)<1 then c:=c+k; fi; od;
%p A210732    if a=b+c then print(n); break; fi;
%p A210732   od;
%p A210732 od; end:
%p A210732 A210732(10000);
%Y A210732 Cf. A066272, A066417, A083207, A204830, A204831, A211223-A211225.
%K A210732 nonn,new
%O A210732 3,1
%A A210732 _Paolo P. Lava_, May 10 2012

%I A212056
%S A212056 0,0,1,19,80,230,521,1019,1800,2966,4593,6819,9768,13566,18353,24339,
%T A212056 31640,40478,51025,63523,78168,95230,114881,137459,163184,192374,
%U A212056 225265,262251,303568,349606,400633,457099,519280,587654,662481
%N A212056 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>y*z+2.
%C A212056 A212055(n)+A212056(n)=n^4.  See A211795 for a guide to related sequences.
%t A212056 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212056 (Do[If[w*x > y*z + 2, s = s + 1],
%t A212056 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212056 Map[t[#] &, Range[0, 50]] (* A212056 *)
%t A212056 (* Peter Moses, Apr 13 2012 *)
%Y A212056 Cf. A211795.
%K A212056 nonn,new
%O A212056 0,4
%A A212056 _Clark Kimberling_, Apr 29 2012

%I A212055
%S A212055 0,1,15,62,176,395,775,1382,2296,3595,5407,7822,10968,14995,20063,
%T A212055 26286,33896,43043,53951,66798,81832,99251,119375,142382,168592,
%U A212055 198251,231711,269190,311088,357675,409367,466422,529296,598267,673855
%N A212055 Number of (w,x,y,z) with all terms in {1,...,n} and w*x<=y*z+2.
%C A212055 A212055(n)+A212056(n)=n^4.  See A211795 for a guide to related sequences.
%t A212055 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212055  (Do[If[w*x <= y*z + 2, s = s + 1],
%t A212055 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212055 Map[t[#] &, Range[0, 50]] (* A212055 *)
%t A212055 (* Peter Moses, Apr 13 2012 *)
%Y A212055 Cf. A211795.
%K A212055 nonn,new
%O A212055 0,3
%A A212055 _Clark Kimberling_, Apr 29 2012

%I A212054
%S A212054 0,0,3,25,98,260,571,1089,1898,3084,4755,7017,9994,13836,18691,24705,
%T A212054 32066,40964,51579,64145,78850,95956,115723,138385,164170,193436,
%U A212054 226467,263521,304930,351076,402195,458777,521082,589532,664547
%N A212054 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>y*z+1.
%C A212054 A212054(n)+A212053(n)=n^4.  See A211795 for a guide to related sequences.
%t A212054 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212054 (Do[If[w*x > y*z + 1, s = s + 1],
%t A212054 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212054 Map[t[#] &, Range[0, 50]] (* A212054 *)
%t A212054 (* Peter Moses, Apr 13 2012 *)
%Y A212054 Cf. A211795.
%K A212054 nonn,new
%O A212054 0,3
%A A212054 _Clark Kimberling_, Apr 29 2012

%I A212053
%S A212053 0,1,13,56,158,365,725,1312,2198,3477,5245,7624,10742,14725,19725,
%T A212053 25920,33470,42557,53397,66176,81150,98525,118533,141456,167606,
%U A212053 197189,230509,267920,309726,356205,407805,464744,527494,596389,671789
%N A212053 Number of (w,x,y,z) with all terms in {1,...,n} and w*x<=y*z+1.
%C A212053 A212053(n)+A212054(n)=n^4.  See A211795 for a guide to related sequences.
%t A212053 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212053 (Do[If[w*x <= y*z + 1, s = s + 1],
%t A212053 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212053 Map[t[#] &, Range[0, 50]] (* A212053 *)
%t A212053 (* Peter Moses, Apr 13 2012 *)
%Y A212053 Cf. A211795.
%K A212053 nonn,new
%O A212053 0,3
%A A212053 _Clark Kimberling_, Apr 29 2012

%I A212066
%S A212066 0,0,4,24,71,166,322,571,928,1403,2043,2875,3906,5183,6727,8584,10762,
%T A212066 13303,16219,19601,23442,27766,32636,38080,44130,50771,58147,66243,
%U A212066 75099,84768,95238,106629,118939,132179,146448,161761,178106
%N A212066 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>x*y*z.
%C A212066 A212066(n)+A212065(n)=n^4.  For a guide to related sequences, see A211795.
%t A212066 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212066 (Do[If[w^2 > x*y*z, s = s + 1],
%t A212066 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212066 Map[t[#] &, Range[0, 50]] (* A212066 *)
%t A212066 (* Peter Moses, Apr 13 2012 *)
%Y A212066 Cf. A211795.
%K A212066 nonn,new
%O A212066 0,3
%A A212066 _Clark Kimberling_, Apr 30 2012

%I A212065
%S A212065 0,1,12,57,185,459,974,1830,3168,5158,7957,11766,16830,23378,31689,
%T A212065 42041,54774,70218,88757,110720,136558,166715,201620,241761,287646,
%U A212065 339854,398829,465198,539557,622513,714762,816892,929637,1053742
%N A212065 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<=x*y*z.
%C A212065 A212065(n)+A212066(n)=n^4.  For a guide to related sequences, see A211795.
%t A212065 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212065 (Do[If[w^2 <= x*y*z, s = s + 1],
%t A212065 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212065 Map[t[#] &, Range[0, 50]] (* A212065 *)
%t A212065 (* Peter Moses, Apr 13 2012 *)
%Y A212065 Cf. A211795.
%K A212065 nonn,new
%O A212065 0,3
%A A212065 _Clark Kimberling_, Apr 30 2012

%I A212064
%S A212064 0,1,8,31,87,185,353,605,978,1471,2123,2958,4028,5308,6864,8733,10947,
%T A212064 13491,16452,19837,23711,28047,32929,38376,44498,51181,58569,66699,
%U A212064 75588,85260,95808,107202,119569,132821,147102,162427,178898
%N A212064 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x*y*z.
%C A212064 A212064(n)+A212063(n)=n^4.  For a guide to related sequences, see A211795.
%t A212064 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212064 (Do[If[w^2 >= x*y*z, s = s + 1],
%t A212064 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212064 Map[t[#] &, Range[0, 50]] (* A212064 *)
%t A212064 (* Peter Moses, Apr 13 2012 *)
%Y A212064 Cf. A211795.
%K A212064 nonn,new
%O A212064 0,3
%A A212064 _Clark Kimberling_, Apr 30 2012

%I A212063
%S A212063 0,0,8,50,169,440,943,1796,3118,5090,7877,11683,16708,23253,31552,
%T A212063 41892,54589,70030,88524,110484,136289,166434,201327,241465,287278,
%U A212063 339444,398407,464742,539068,622021,714192,816319,929007,1053100
%N A212063 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<x*y*z.
%C A212063 A212063(n)+A212064(n)=n^4.  For a guide to related sequences, see A211795.
%t A212063 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212063 (Do[If[w^2 < x*y*z, s = s + 1],
%t A212063 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212063 Map[t[#] &, Range[0, 50]] (* A212063 *)
%t A212063 (* Peter Moses, Apr 13 2012 *)
%Y A212063 Cf. A211795.
%K A212063 nonn,new
%O A212063 0,3
%A A212063 _Clark Kimberling_, Apr 30 2012

%I A212059
%S A212059 0,0,3,9,12,21,24,34,40,49,52,70,73,82,91,106,109,127,130,148,157,166,
%T A212059 169,199,205,214,224,242,245,272,275,296,305,314,323,359,362,371,380,
%U A212059 410,413,440,443,461,479,488,491,536,542,560,569,587,590,620
%N A212059 Number of (w,x,y,z) with all terms in {1,...,n} and w=x*y*z-1.
%C A212059 For a guide to related sequences, see A211795.
%t A212059 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212059 (Do[If[w == x*y*z - 1, s = s + 1],
%t A212059 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212059 Map[t[#] &, Range[0, 60]] (* A212059 *)
%t A212059 (* Peter Moses, Apr 13 2012 *)
%Y A212059 Cf. A211795.
%K A212059 nonn,new
%O A212059 0,3
%A A212059 _Clark Kimberling_, Apr 30 2012

%I A212352
%S A212352 0,1,3,9,25,75,235,759,2521,8555,29503,103129,364547,1300819,4679471,
%T A212352 16952161,61790441,226451035,833918839,3084255127,11451630043,
%U A212352 42669225171,159497648599,597950875255,2247724108771,8470205600639
%N A212352 Row sums of A047997.
%Y A212352 Equals A047653(n) - 1. Cf. A047997.
%K A212352 nonn,new
%O A212352 0,3
%A A212352 _N. J. A. Sloane_, May 16 2012

%I A212273
%S A212273 1,2,12,108,1240,16980,267624,4750872,93615408,2026018980,47764893000,
%T A212273 1218621868200,33459133580496,983950866169544,30862364045494800,
%U A212273 1028656519404191280,36312294166359724128,1353561646509338629572,53133041579508472510728
%N A212273 a(n) = n * A000699(n).
%F A212273 a(n) = [x^n] A178685(x)^n.
%e A212273 x + 2*x^2 + 12*x^3 + 108*x^4 + 1240*x^5 + 16980*x^6 + 267624*x^7 + 4750872*x^8 + ...
%o A212273 (PARI) {a(n) = local(A); A = O(x) ; for( i=1, n, A = x + A * (2 * x * A' - A)); n * polcoeff(A, n)}
%Y A212273 Cf. A000699, A178685.
%K A212273 nonn,new
%O A212273 1,2
%A A212273 _Michael Somos_, May 12 2012

%I A212409
%S A212409 2,4,8,27,54,156,380,476,782,861,1053,1976,2542,2565,3125,3213,3368,
%T A212409 6250,8732,13338,13724,22734,42716,46136,51640,56156,56444,58941,
%U A212409 64796,67196,92637,115198,121875,251516,261598,288333,296875,311418,348570,371875,379053
%N A212409 Number n for which n, n' and n'' are in arithmetic progression, where n' and n'' are the first and second arithmetic derivatives.
%C A212409 A051674 is a subset of this sequence.
%C A212409 If we consider also the third derivative, the numbers for which stand n'''-n''=n''-n'=n'-n are 4, 27, 380, 2565, 3125, 296875, 696764, 823543, …
%H A212409 Donovan Johnson, <a href="/A212409/b212409.txt">Table of n, a(n) for n = 1..300</a>
%e A212409 n=8732, n'=9116, n''=9500,  8732-9116=9116-9500=-384
%e A212409 n=115198, n'=58559, n''=1920, 115198-58559=58559-1920=56639
%p A212409 with(numtheory);
%p A212409 A212409:= proc(n)
%p A212409 local a,b,i,p,pfs;
%p A212409 for i from 1 to n do
%p A212409   pfs:=ifactors(i)[2];  a:=i*add(op(2,p)/op(1,p),p=pfs) ;
%p A212409   pfs:=ifactors(a)[2];  b:=a*add(op(2,p)/op(1,p),p=pfs) ;
%p A212409   if b-a=a-i then print(i); fi;
%p A212409 od;
%p A212409 end:
%p A212409 A212409(1000000);
%Y A212409 Cf. A003415
%K A212409 nonn,new
%O A212409 1,1
%A A212409 _Paolo P. Lava_, May 15 2012

%I A212422
%S A212422 1,2,4,11,30,79,212,574,1588,4455,12478,35377,100335,285385,812648,
%T A212422 2318781,6620447,18935754,54209820,155482355,446673973,1284474758,
%U A212422 3698559438,10659129412,30740979089,88719378807,256240088606,740502848865,2141552451104,6197576727082
%N A212422 Sums involving triangle A096815.
%F A212422 S(n,k) = sum(k=0..n, T(n,k) * T(n+1,k+1) ), where T(n,k) = A096815(n,k).
%t A212422 T[n_,k_] := T[n,k] = If[n < k || k < 0, 0, If[k = 1 || k == n, 1, Sum[T[n-k,j]T[k,k-j],{j,0,k}]]]
%t A212422 Table[Sum[T[n,k]T[n+1,k+1],{k,0,n}],{n,0,50}]
%o A212422 (Maxima) T(n,k):= if ( n<k or k<0 ) then 0 else
%o A212422 if ( k<=1 or k=n ) then 1 else sum(T(n-k,j)*T(k,k-j), j, 0, k);
%o A212422 makelist(sum(T(n,k)*T(n+1,k+1),k,0,n),n,0,14);
%Y A212422 Cf. A096815, A096816, A212264, A212265, A212421.
%K A212422 nonn,new
%O A212422 0,2
%A A212422 _Emanuele Munarini_, May 16 2012

%I A212421
%S A212421 1,1,3,6,17,42,118,320,861,2442,6658,18686,52100,146832,410990,
%T A212421 1161722,3282403,9283968,26336598,74712158,212613094,605118528,
%U A212421 1726261816,4926677982,14079210098,40247549662,115175011608,329707161766,944566633454,2707713181042
%N A212421 Central coefficients of triangle A096815.
%F A212421 a(n) = T(2n,n), where T(n,k) = A096815(n,k).
%t A212421 T[n_,k_] := T[n,k] = If[n < k || k < 0, 0, If[k = 1 || k == n, 1, Sum[T[n-k,j]T[k,k-j],{j,0,k}]]]
%t A212421 Table[T[2n, n], {n, 0, 50}]
%Y A212421 Cf. A096815, A096816, A212264, A212265, A212422
%K A212421 nonn,new
%O A212421 0,3
%A A212421 _Emanuele Munarini_, May 16 2012

%I A212265
%S A212265 1,2,3,7,21,51,138,358,969,2735,7514,21372,60144,171510,485294,
%T A212265 1388586,3949211,11296953,32294750,92471813,265484354,762548923,
%U A212265 2194099432,6318266076,18214160616,52525201261,151638570041,437959067720,1265890334086,3661612407681
%N A212265 Row square-sums of triangle A096815.
%t A212265 T[n_,k_] := T[n,k] = If[n < k || k < 0, 0, If[k = 1 || k == n, 1, Sum[T[n-k,j]T[k,k-j],{j,0,k}]]]
%t A212265 Table[Sum[T[n,k]^2,{k,0,n}],{n,0,50}]
%o A212265 (Maxima) T(n,k):= if ( n<k or k<0 ) then 0 else
%o A212265 if ( k<=1 or k=n ) then 1 else sum(T(n-k,j)*T(k,k-j),j,0, k);
%o A212265 makelist(sum(T(n,k)^2,k,0,n),n,0,20);
%Y A212265 Cf. A096815, A096816, A212264, A212421, A212422.
%K A212265 nonn,new
%O A212265 0,2
%A A212265 _Emanuele Munarini_, May 12 2012

%I A182435
%S A182435 0,1,4,21,120,697,4060,23661,137904,803761,4684660,27304197,159140520,
%T A182435 927538921,5406093004,31509019101,183648021600,1070379110497,
%U A182435 6238626641380,36361380737781,211929657785304,1235216565974041,7199369738058940,41961001862379597
%N A182435 a(n) = 6*a(n-1) - a(n-2) - 2 with n>1, a(0)=0, a(1)=1.
%C A182435 If p equals a prime of the form 8*r +/- 3 then a(p + 1) == 0 (mod p); if p is a prime in the form of 8*r +/- 1 then a(p + 1) == 4 (mod p).
%H A182435 Bruno Berselli, <a href="/A182435/b182435.txt">Table of n, a(n) for n = 0..100</a>
%H A182435 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182435 a(n) = 1/2+1/4*sqrt(2)*((3+2*sqrt(2))^n-(3-2*sqrt(2))^n)-1/4*((3-2*sqrt(2))^n+(3+2*sqrt(2))^n). - _Paolo P. Lava_, May 10 2012
%F A182435 G.f.: x*(1-3*x)/((1-x)*(1-6*x+x^2)). - _Bruno Berselli_, May 15 2012
%F A182435 a(n) = A001652(n-1)+1 with A001652(-1)=-1. - _Bruno Berselli_, May 16 2012
%t A182435 m = -20;
%t A182435 n = -3;
%t A182435 c = 0;
%t A182435 list3 = Reap[While[c < 20,t = 6 n - m - 2;Sow[t];m = n;n = t; c++]][[2,1]]
%o A182435 (MAGMA) [n le 2 select n-1 else 6*Self(n-1)-Self(n-2)-2: n in [1..24]]; // _Bruno Berselli_, May 15 2012
%Y A182435 Cf. A001108.
%K A182435 nonn,easy,new
%O A182435 0,3
%A A182435 _Kenneth J Ramsey_, Apr 28 2012

%I A182191
%S A182191 1,5,43,265,1559,9101,53059,309265,1802543,10506005,61233499,
%T A182191 356895001,2080136519,12123924125,70663408243,411856525345,
%U A182191 2400475743839,13990997937701,81545511882379,475282073356585,2770146928257143,16145599496186285,94103450048860579
%V A182191 -1,5,43,265,1559,9101,53059,309265,1802543,10506005,61233499,
%W A182191 356895001,2080136519,12123924125,70663408243,411856525345,
%X A182191 2400475743839,13990997937701,81545511882379,475282073356585,2770146928257143,16145599496186285,94103450048860579
%N A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.
%C A182191 If p is a prime of the form 8*r +/- 3 then a(p) == 1 (mod p); if p is a prime of the form 8*r +/- 1 then a(p) == 5 (mod p).
%H A182191 Bruno Berselli, <a href="/A182191/b182191.txt">Table of n, a(n) for n = 0..100</a>
%H A182191 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182191 G.f.: -(1-12*x-x^2)/((1-x)*(1-6*x+x^2)). [_Bruno Berselli_, May 15 2012]
%F A182191 a(n) = 2*A038723(n)-3. [_Bruno Berselli_, May 16 2012]
%t A182191 m = 19;n = 1; c = 0;
%t A182191 list3 = Reap[While[c < 22, t = 6 n - m + 12; Sow[t];m = n; n = t;c++]][[2,1]]
%o A182191 (MAGMA) I:=[-1,5]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..19]]; // _Bruno Berselli_, May 15 2012
%Y A182191 Cf. A038723.
%K A182191 sign,easy,new
%O A182191 0,2
%A A182191 _Kenneth J Ramsey_, Apr 17 2012

%I A212424
%S A212424 4181,5777,6721,10877,13201,15251,34561,51841,64079,64681,67861,68251,
%T A212424 75077,90061,96049,97921,100127,113573,118441,146611,161027,162133,
%U A212424 163081,186961,197209,219781,231703,252601,254321,257761,268801,272611
%N A212424 Frobenius pseudoprimes with respect to Fibonacci polynomial x^2-x-1.
%C A212424 Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2-x-1 is 5777". Crandall and Pomerance state that the first such Frobenius pseudoprime is actually 4181.
%D A212424 R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.
%H A212424 Weisstein, Eric W. <a href="http://mathworld.wolfram.com/FrobeniusPseudoprime.html">Frobenius Pseudoprime</a>, MathWorld.
%H A212424 Jon Grantham (2001). Frobenius pseudoprimes. Mathematics of Computation 70 (234): 873-891. doi:<a href="http://dx.doi.org/10.1090/S0025-5718-00-01197-2">10.1090/S0025-5718-00-01197-2</a>
%H A212424 Max Alekseyev, <a href="/A212424/b212424.txt">Table of n, a(n) for n = 1..653</a>
%o A212424 (PARI) { isFP(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x)||(kronecker(5,n)==1 && t==x) }
%Y A212424 Cf. A212423, A005845, A094395, A094063, A094411
%K A212424 nonn,new
%O A212424 1,1
%A A212424 _Max Alekseyev_, May 16 2012

%I A212423
%S A212423 5777,10877,75077,100127,113573,161027,162133,231703,430127,635627,
%T A212423 851927,1033997,1106327,1256293,1388903,1697183,2263127,2435423,
%U A212423 2662277,3175883,3399527,3452147,3774377,3900797,4109363,4226777,4403027,4828277,4870847
%N A212423 Frobenius pseudoprimes == 2,3 (mod 5) with respect to Fibonacci polynomial x^2-x-1.
%C A212423 Grantham incorrectly claims that "the first Frobenius pseudoprime with respect to the Fibonacci polynomial x^2-x-1 is 5777". However n=5777 is the first Frobenius pseudoprime with respect to x^2-x-1 that has jacobi symbol (5/n)=-1, i.e., n==2,3 (mod 5). Unrestricted version with the first term 4181 is given in A212424.
%C A212423 Intersection of A212424 and A047221.
%D A212423 R. Crandall, C. B. Pomerance. Prime Numbers: A Computational Perspective. Springer, 2nd ed., 2005.
%H A212423 Weisstein, Eric W. <a href="http://mathworld.wolfram.com/FrobeniusPseudoprime.html">Frobenius Pseudoprime</a>, MathWorld.
%H A212423 Jon Grantham (2001). Frobenius pseudoprimes. Mathematics of Computation 70 (234): 873-891. doi:<a href="http://dx.doi.org/10.1090/S0025-5718-00-01197-2">10.1090/S0025-5718-00-01197-2</a>
%o A212423 (PARI) { isFP23(n) = if(ispseudoprime(n),return(0)); t=Mod(x*Mod(1,n),(x^2-x-1)*Mod(1,n))^n; (kronecker(5,n)==-1 && t==1-x) }
%Y A212423 Cf. A094395, A094063, A094411
%K A212423 nonn,new
%O A212423 1,1
%A A212423 _Max Alekseyev_, May 16 2012

%I A182287
%S A182287 1,1,2,6,24,120,720,5040,40320,362880,11,11,12,16,34,130,730,5050,
%T A182287 40330,362890,21,21,22,26,44,140,740,5060,40340,362900,61,61,62,66,84,
%U A182287 180,780,5100,40380,362940,241,241,242,246,264,360,960,5280,40560,363120
%N A182287 If n = p*10^i + q*10^(i-1) + r*10^(i-2) + ... in decimal notation, then a(n) = p!*10^i + q!*10^(i-1) + r!*10^(i-2)+ ... .
%H A182287 Bruno Berselli, <a href="/A182287/b182287.txt">Table of n, a(n) for n = 0..1000</a>
%H A182287 Renzo Remotti, <a href="/A182287/a182287.pdf">Positional Factorial Sequence</a>
%e A182287 a(1)=1 because 1!*10^0=1, a(15)=130 because 1!*10^1+5!*10^0=130.
%o A182287 (MAGMA) [n eq 0 select 1 else &+[Factorial(Reverse(Intseq(n))[k])*10^(#Intseq(n)-k): k in [1..#Intseq(n)]]: n in [0..50]]; // _Bruno Berselli_, May 15 2012
%Y A182287 Cf. A007623, A061602.
%K A182287 nonn,base,new
%O A182287 0,3
%A A182287 _Renzo Remotti_, Apr 23 2012
%E A182287 Offset changed from 1 to 0 by _Bruno Berselli_, May 16 2012

%I A212264
%S A212264 1,1,2,2,3,4,6,9,12,18,26,37,52,77,110,156,226,324,464,668,960,1370,
%T A212264 1977,2834,4060,5832,8371,11993,17219,24695,35408,50837,72880,104619,
%U A212264 150065,215343,308868,443380,635753,912581,1308771,1878238,2693939,3866059,5544795
%N A212264 Diagonal sums of triangle A096815.
%t A212264 T[n_,k_] := T[n,k] = If[n < k || k < 0, 0, If[k = 1 || k == n, 1, Sum[T[n-k,j]T[k,k-j], {j,0,k}]]]
%t A212264 Table[Sum[T[n-k,k],{k,0,n}],{n,0,50}]
%o A212264 (Maxima) T(n,k):= if ( n<k or k<0 ) then 0 else
%o A212264 if ( k<=1 or k=n ) then 1 else sum(T(n-k,j)*T(k,k-j),j,0, k);
%o A212264 makelist(sum(T(n-k,k),k,0,n/2),n,0,32);
%Y A212264 Cf. A096815, A096816.
%K A212264 nonn,new
%O A212264 0,3
%A A212264 _Emanuele Munarini_, May 12 2012

%I A212020
%S A212020 0,0,0,8,14,18,48,56,74,126,154,166,262,278,318,422,480,500,674,698,
%T A212020 800,956,1024,1052,1308,1378,1458,1698,1852,1888,2240,2280,2434,2686,
%U A212020 2794,2950,3472,3520,3640,3944,4234,4286,4800,4856,5102,5672
%N A212020 Number of (w,x,y,z) with all terms in {1,...,n} and w*x=3*y*z.
%C A212020 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212020 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212020 (Do[If[w*x == 3 y*z, s = s + 1],
%t A212020 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212020 Map[t[#] &, Range[0, 50]] (* A212020 *)
%t A212020 %/2  (* integers *)
%t A212020 (* Peter Moses, Apr 13 2012 *)
%Y A212020 Cf. A211795.
%K A212020 nonn,new
%O A212020 0,4
%A A212020 _Clark Kimberling_, Apr 28 2012

%I A212060
%S A212060 0,0,3,6,15,18,28,34,43,46,64,67,76,85,100,103,121,124,142,151,160,
%T A212060 163,193,199,208,218,236,239,266,269,290,299,308,317,353,356,365,374,
%U A212060 404,407,434,437,455,473,482,485,530,536,554,563,581,584,614,623
%N A212060 Number of (w,x,y,z) with all terms in {1,...,n} and w=x*y*z-2.
%C A212060 For a guide to related sequences, see A211795.
%t A212060 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212060 (Do[If[w == x*y*z - 2, s = s + 1],
%t A212060 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212060 Map[t[#] &, Range[0, 60]] (* A212060 *)
%t A212060 (* Peter Moses, Apr 13 2012 *)
%Y A212060 Cf. A211795.
%K A212060 nonn,new
%O A212060 0,3
%A A212060 _Clark Kimberling_, Apr 30 2012

%I A212061
%S A212061 0,1,3,5,12,14,18,20,30,45,49,51,65,67,71,75,106,108,134,136,150,154,
%T A212061 158,160,180,215,219,241,255,257,265,267,307,311,315,319,392,394,398,
%U A212061 402,422,424,432,434,448,478,482,484,540,599,657,661,675,677
%N A212061 Number of (w,x,y,z) with all terms in {1,...,n} and w*x=(y*z)^2.
%C A212061 For a guide to related sequences, see A211795.
%t A212061 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212061 (Do[If[w*x == (y*z)^2, s = s + 1],
%t A212061 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212061 Map[t[#] &, Range[0, 50]] (* A212061 *)
%t A212061 (* Peter Moses, Apr 13 2012 *)
%Y A212061 Cf. A211795.
%K A212061 nonn,new
%O A212061 0,3
%A A212061 _Clark Kimberling_, Apr 30 2012

%I A212058
%S A212058 0,1,5,12,25,41,66,94,132,176,229,285,359,436,522,617,727,840,971,
%T A212058 1105,1257,1418,1588,1761,1964,2173,2391,2619,2865,3114,3390,3669,
%U A212058 3969,4278,4596,4923,5286,5652,6027,6411,6825,7242,7686,8133,8598
%N A212058 Number of (w,x,y,z) with all terms in {1,...,n} and w>=x*y*z.
%C A212058 A212058(n)+A212057(n)=n^4.  For a guide to related sequences, see A211795.
%t A212058 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212058 (Do[If[w >= x*y*z, s = s + 1],
%t A212058 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212058 Map[t[#] &, Range[0, 50]] (* A212058 *)
%t A212058 (* Peter Moses, Apr 13 2012 *)
%Y A212058 Cf. A211795, A061201.
%K A212058 nonn,new
%O A212058 0,3
%A A212058 _Clark Kimberling_, Apr 30 2012

%I A212057
%S A212057 0,0,11,69,231,584,1230,2307,3964,6385,9771,14356,20377,28125,37894,
%T A212057 50008,64809,82681,104005,129216,158743,193063,232668,278080,329812,
%U A212057 388452,454585,528822,611791,704167,806610,919852,1044607,1181643
%N A212057 Number of (w,x,y,z) with all terms in {1,...,n} and w<x*y*z.
%C A212057 A212057(n)+A212058(n)=n^4.  For a guide to related sequences, see A211795.
%t A212057 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212057 (Do[If[w < x*y*z, s = s + 1],
%t A212057 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212057 Map[t[#] &, Range[0, 50]] (* A212057 *)
%t A212057 (* Peter Moses, Apr 13 2012 *)
%Y A212057 Cf. A211795, A061201.
%K A212057 nonn,new
%O A212057 0,3
%A A212057 _Clark Kimberling_, Apr 30 2012

%I A212062
%S A212062 0,1,4,7,16,19,31,34,50,68,80,83,122,125,137,149,185,188,233,236,269,
%T A212062 281,293,296,368,410,422,456,489,492,570,573,630,642,654,666,792,795,
%U A212062 807,819,891,894,954,957,990,1047,1059,1062,1191,1269,1350
%N A212062 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x*y*z.
%C A212062 For a guide to related sequences, see A211795.
%t A212062 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212062 (Do[If[w^2 == x*y*z, s = s + 1],
%t A212062 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212062 Map[t[#] &, Range[0, 50]] (* A212062 *)
%t A212062 (* Peter Moses, Apr 13 2012 *)
%Y A212062 Cf. A211795.
%K A212062 nonn,new
%O A212062 0,3
%A A212062 _Clark Kimberling_, Apr 30 2012

%I A212393
%S A212393 1,1,2,5,14,30,72,195,485,1059,2065,3682,6120,9620,14454,20925,29367,
%T A212393 40145,53655,70324,90610,115002,144020,178215,218169,264495,317837,
%U A212393 378870,448300,526864,615330,714497,825195,948285,1084659,1235240,1400982,1582870,1781920
%N A212393 Expansion of (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
%C A212393 In the paper of Kitaev, Remmel and Tiefenbruck (see Link lines), Q_(132)^(0,k,0,0)(x,0) represents a generating function depending on k and x.
%C A212393 For successive values of k we have:
%C A212393 k=1, the g.f. of A000012: 1/(1-x);
%C A212393 k=2, the g.f. of A028310: (1-x+x^2)/(1-x)^2;
%C A212393 k=3, the g.f. (1-2*x+2*x^2+x^3-x^4)/(1-x)^3, whose coefficients (except the first two) are given by A000096 (for n>0);
%C A212393 k=4, the g.f. (1-3*x+4*x^2-x^3+3*x^4-5*x^5+2*x^6)/(1-x)^4, whose coefficients (except the first three) are given by A005586 (for n>0).
%C A212393 This sequence corresponds to the case k=5.
%H A212393 Bruno Berselli, <a href="/A212393/b212393.txt">Table of n, a(n) for n = 0..1000</a>
%H A212393 Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012 (page 21, Theorem 10).
%H A212393 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A212393 G.f.: (1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5.
%F A212393 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>8, a(0)=a(1)=1, a(2)=2, a(3)=5, a(4)=14, a(5)=30, a(6)=72, a(7)=195, a(8)=485.
%F A212393 a(n) = (n-3)*(31*n^3-369*n^2+1454*n-1560)/24 for n>3, a(0)=a(1)=1, a(2)=2, a(3)=5.
%t A212393 CoefficientList[Series[(1 - 4 x + 7 x^2 - 5 x^3 + 4 x^4 - 6 x^5 + 21 x^6 + 18 x^7 - 5 x^8)/(1 - x)^5, {x, 0, 38}], x]
%o A212393 (PARI) Vec((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5+O(x^39))
%o A212393 (MAGMA) m:=39; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x+7*x^2-5*x^3+4*x^4-6*x^5+21*x^6+18*x^7-5*x^8)/(1-x)^5));
%K A212393 nonn,easy,new
%O A212393 0,3
%A A212393 _Bruno Berselli_, May 14 2012

%I A212207
%S A212207 1,1,1,1,3,2,1,6,9,4,1,10,26,25,8
%N A212207 Triangle read by rows: coefficients of polynomials p_{n,n-1}(x) arising in enumeration of two-line arrays.
%C A212207 These polynomials are defined in Section 3 of Carlitz-Riordan (1971). Equation (3.14) claims to be a recurrence, which unfortunately I could not get to work. The coefficients of the polynomials A_n(x) = a_{n,n}(x) which appear in (3.14) are the Narayama numbers A001263.
%D A212207 Carlitz, L.; Riordan, John. Enumeration of some two-line arrays by extent. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66).
%e A212207 Triangle begins:
%e A212207 1
%e A212207 1 1
%e A212207 1 3 2
%e A212207 1 6 9 4
%e A212207 1 10 26 25 8
%e A212207 ...
%Y A212207 Cf. A001263.
%K A212207 nonn,tabl,more,new
%O A212207 0,5
%A A212207 _N. J. A. Sloane_, May 15 2012

%I A212351
%S A212351 6,8,12,24,40,80,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A212351 131072,262144,524288,1048576
%N A212351 Maximal number of "good" manifolds in an n-nice polytope.
%D A212351 Altshuler, Amos. Manifolds in stacked 4-polytopes. J. Combinatorial Theory Ser. A 10 1971 198--239. MR0283688 (44 #918)
%F A212351 2^n for n >= 7.
%Y A212351 Cf. A212350.
%K A212351 nonn,new
%O A212351 1,1
%A A212351 _N. J. A. Sloane_, May 15 2012

%I A212350
%S A212350 6,8,12,20,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,
%T A212350 131072,262144,524288,1048576
%N A212350 Maximal number of "good" manifolds in an n-serial polytope.
%D A212350 Altshuler, Amos. Manifolds in stacked 4-polytopes. J. Combinatorial Theory Ser. A 10 1971 198--239. MR0283688 (44 #918)
%F A212350 2^n for n >= 5.
%Y A212350 Cf. A212351.
%K A212350 nonn,new
%O A212350 1,1
%A A212350 _N. J. A. Sloane_, May 15 2012

%I A212413
%S A212413 1,1,2,9,63
%N A212413 Anchored partitions of a circle
%C A212413 n line segments are drawn successively within a circle; they may not cross one another. When each segment is drawn, each of its endpoints must be "anchored"; that is, it must lie either on the circumference of the circle, or on a previously drawn segment. No two endpoints may coincide (thus no "V"s or "X"s). The sequence counts the topologically distinct partitions, and does not count separately partitions that are equivalent under mirror reflection.
%H A212413 Jon Wild, <a href="/A212413/a212413.pdf">Illustration for a(3)=9 and a(4)=63</a>
%e A212413 In the attached pdf file, the nine anchored partitions for n=3 are shown in the left-hand margin. For each, all partitions for n=4 are illustrated that can be derived from the n=3 cases by adding one line segment, except those that have already been derived from an earlier n=3 case.
%K A212413 nonn,new
%O A212413 0,3
%A A212413 _Jon Wild_, May 15 2012

%I A212411
%S A212411 1,1,2,7,36,235,1792,15261,141382,1401334,14694166,161714217,
%T A212411 1857003186,22152227989,273573165626,3488210643709,45820081884234,
%U A212411 618950367384072,8585324020132250,122127635117014779,1779763238159032068,26545963246376545934
%N A212411 G.f. satisfies: A(x) = 1 + x*A(1 - 1/A(x))^2.
%C A212411 Compare g.f. to the identity: G(x) = 1 + x*G(1-1/G(x)) when G(x) = 1/(1-x).
%H A212411 Vincenzo Librandi, <a href="/A212411/b212411.txt">Table of n, a(n) for n = 0..100</a>
%F A212411 Given g.f. A(x), let G(x) be the g.f. of unsigned A139702, then:
%F A212411 (1) G(x) = x*A(G(x)^2/x),
%F A212411 (2) A( x/(1 - G(x)^2/x) ) = 1/(1 - G(x)^2/x),
%F A212411 (3) x = G(x - G(x)^2).
%F A212411 Given g.f. A(x), let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)) and A(x) = 1 + x*F(x)^2.
%e A212411 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 +...
%e A212411 Related expansions:
%e A212411 A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 90*x^4 + 570*x^5 + 4247*x^6 +...
%e A212411 1 - 1/A(x) = x + x^2 + 4*x^3 + 23*x^4 + 161*x^5 + 1286*x^6 + 11321*x^7 +...
%e A212411 A(1-1/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 98*x^4 + 753*x^5 + 6471*x^6 +...
%e A212411 Let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)):
%e A212411 F(1-1/A(x)) = 1 + x + 4*x^2 + 25*x^3 + 193*x^4 + 1693*x^5 + 16240*x^6 +...
%e A212411 ...
%e A212411 Let G(x) be the g.f. of A139702 (unsigned), then
%e A212411 G(x) satisfies: x = G(x - G(x)^2) and G(x) = A(G(x)^2/x), where:
%e A212411 G(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
%e A212411 G(x)^2/x = x + 2*x^2 + 9*x^3 + 56*x^4 + 420*x^5 + 3572*x^6 +...
%e A212411 1/(1-G(x)^2/x) = 1 + x + 3*x^2 + 14*x^3 + 85*x^4 + 616*x^5 + 5072*x^6 +...
%e A212411 such that A(x/(1 - G(x)^2/x)) = 1/(1 - G(x)^2/x).
%o A212411 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*subst(A^2,x,1-1/(A+x*O(x^n))));polcoeff(A,n)}
%o A212411 for(n=0,30,print1(a(n),", "))
%Y A212411 Cf. A139702.
%K A212411 nonn,new
%O A212411 0,3
%A A212411 _Paul D. Hanna_, May 15 2012

%I A212206
%S A212206 1,0,1,0,2,0,1,4,0,0,12,2,0,0,12,2,0,0,12,30,0,0,4,100,28,0,0,0,140,
%T A212206 280,9,0,0,0,90,980,360,0,0,0,22,1680,2940,220
%N A212206 Irregular triangle read by rows: T(n,k) = number of "pat" permutations of [1..n] with k descents.
%C A212206 Row sums are Catalan numbers A000108.
%D A212206 D. Callan, Flexagons lead to a Catalan number identity, Amer. Math. Monthly, 119 (May 2012), 415-419.
%F A212206 T(n,k)=binomial(2n-2k-1,k)*binomial(2k,n-k-1)/(2n-2k-1).
%e A212206 Triangle begins:
%e A212206 1
%e A212206 0 1
%e A212206 0 2
%e A212206 0 1 4
%e A212206 0 0 12 2
%e A212206 0 0 12 2
%e A212206 0 0 12 30
%e A212206 0 0 4 100 28
%e A212206 0 0 0 140 280 9
%e A212206 0 0 0 90 980 360
%e A212206 0 0 0 22 1680 2940 220
%e A212206 ...
%Y A212206 Cf. A000108.
%K A212206 nonn,tabf,more,new
%O A212206 1,5
%A A212206 _N. J. A. Sloane_, May 15 2012

%I A212275
%S A212275 1,2,3,1,5,24,7,8,1,10,99,12,13,56,135,4,425,8,4275,5,84,352,368,6,1,
%T A212275 234,12,28,116,120,124,2,33,306,315,4,37,3800,156,10,6929,42,1075,176,
%U A212275 45,184,47,3,1,2,204,117,1908,6,55,14,1425,58,236,60,10309,62
%N A212275 Least k such that 4*n*k+1 is a prime of the form m^2+1, or 0 if no such k exists.
%C A212275 Conjecture: a(n)>0. If the conjecture is true, then there exist infinitely many primes of the form m^2+1.
%H A212275 Alois P. Heinz and Charles R Greathouse IV, <a href="/A212275/b212275.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Heinz)
%p A212275 a:= proc(n) local k;
%p A212275       for k while not(isprime(4*n*k+1) and issqr (n*k)) do od; k
%p A212275     end:
%p A212275 seq (a(n), n=1..70);  # _Alois P. Heinz_, May 13 2012
%o A212275 (PARI) a(n)=my(N=4*n*core(n),k=0);while(!isprime(k++^2*N+1),);k^2*N/(4*n) \\ _Charles R Greathouse IV_, May 14 2012
%o A212275 (MAGMA) S:=[]; for n in [1..62] do k:=1; while not IsPrime(4*n*k+1) or not IsSquare(n*k) do k:=k+1; end while; Append(~S, k); end for; S; // _Bruno Berselli_, May 15 2012
%Y A212275 Cf. A002496.
%K A212275 nonn,easy,new
%O A212275 1,2
%A A212275 _Vladimir Shevelev_, May 13 2012

%I A211075
%S A211075 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
%T A211075 103,109,113,127,131,139,151,157,163,167,173,181,193,199,211,233,241,
%U A211075 251,257,263,271,283,293,317,331,337,353,359,367,373,383,389,401,409
%N A211075 Primes p followed by prime gap of length log(p/log(p)^2)^2.
%C A211075 Primes followed by unusually long prime gaps.
%H A211075 Charles R Greathouse IV, <a href="/A211075/b211075.txt">Table of n, a(n) for n = 1..4139</a>
%o A211075 (PARI) G=1;p=3;forprime(q=5,1e7,if(q-p>=G,G=log(p/log(p)^2)^2; if(q-p>=G, print1(p", ")));p=q)
%Y A211075 Cf. A211073, A002386, A082885, A111943, A182315.
%K A211075 nonn,new
%O A211075 1,1
%A A211075 _Charles R Greathouse IV_, May 06 2012

%I A211199
%S A211199 0,1,65536,43046721,4294967296,152587890625,2821109907456,
%T A211199 33232930569601,281474976710656,1853020188851841,1,2,65537,43046722,
%U A211199 4294967297,152587890626,2821109907457,33232930569602,281474976710657,1853020188851842,65536,65537,131072,43112257
%N A211199 Sum of the 16-th powers of the decimal digits of n.
%C A211199 This is to exponent 16 as A007953 is to exponent 0, A003132 is to exponent 2, and A055013 is to exponent 4.
%e A211199 a(14) = 1^16 + 4^16 = 4294967297 = 641 * 6700417.
%o A211199 (PARI) a(n)=my(s);while(n,s+=(n%10)^16;n\=10);s \\ _Charles R Greathouse IV_, May 11 2012
%Y A211199 Cf. A007953, A003132, A055013.
%K A211199 nonn,base,easy,new
%O A211199 0,3
%A A211199 _Jonathan Vos Post_, May 11 2012

%I A210937
%S A210937 4,2,1,4,7,8,1,6,1,2,9,8,8,6,7,3,0,9,0,6,2,0,0,9,1,1,0,4,1,1,2,1,3,6,
%T A210937 4,3,1,1,1,4,6,0,3,3,5,0,7,7,6,8,0,9,0,3,9,6,8,4,3,3,7,4,7,8,7,9,0,8,
%U A210937 7,9,1,4,5,4,0,0,2,2,2,0,4,8,8,1,6,9,0,0,8,5,8,7,0,5,4,9,6,8,4,4,7,5,3,5,8,2,8,2,4,3,0,7,7,2,5,0,5,0,2,4,2,5,4,2,5,8,2,8,2
%N A210937 Decimal expansion of the continued fraction 1'+1/(2'+2/(3'+3/...)), where n' is the arithmetic derivative of n.
%C A210937 A good approximation up to the 9th decimal digit is 4796/11379.
%D A210937 1
%e A210937 0.42147816129886730906200911...
%p A210937 with(numtheory);
%p A210937 A210937:= proc(n)
%p A210937 local a,b,c,I,p,pfs;
%p A210937 b:=1;
%p A210937 for i from n by -1 to 2 do
%p A210937   pfs:=ifactors(i)[2]; a:=i*add(op(2,p)/op(1,p),p=pfs); b:=1/b*a+i;
%p A210937 od;
%p A210937 print(evalf(b,500));
%p A210937 end:
%p A210937 A210937(10000);
%Y A210937 Cf. A003415, A190144, A190145, A190146, A190147, A209873.
%K A210937 nonn,cons,new
%O A210937 0,1
%A A210937 _Paolo P. Lava_, May 11 2012

%I A212408
%S A212408 55,285,1314,5769,24322,100736,413220,1685039,6844362,27724036,
%T A212408 112072540,452348578,1823583124,7344493104,29556979016,118871913787,
%U A212408 477820811258,1919788147772,7710323488748,30956089143902,124248950086268
%N A212408 Number of binary arrays of length 2*n+6 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212408 Row 7 of A212402
%H A212408 R. H. Hardin, <a href="/A212408/b212408.txt">Table of n, a(n) for n = 1..210</a>
%e A212408 Some solutions for n=3
%e A212408 ..0....0....0....1....1....0....1....0....0....1....0....1....1....0....1....1
%e A212408 ..0....0....1....0....0....0....0....0....1....0....1....0....0....0....0....0
%e A212408 ..0....1....0....0....0....0....1....1....1....0....1....0....1....0....1....0
%e A212408 ..1....0....1....1....1....0....0....1....0....0....0....1....0....1....0....1
%e A212408 ..0....0....0....0....1....0....1....0....0....1....0....0....0....0....0....1
%e A212408 ..0....0....0....0....0....0....0....0....0....0....0....1....0....0....0....0
%e A212408 ..0....0....0....1....1....1....0....1....1....0....0....0....1....1....0....0
%e A212408 ..0....1....0....0....0....0....0....0....0....0....1....1....1....0....0....0
%e A212408 ..1....0....0....0....0....1....1....1....0....0....1....0....0....0....1....0
%e A212408 ..1....0....0....1....1....0....1....1....0....0....0....1....0....1....0....0
%e A212408 ..0....1....0....0....0....0....0....0....1....1....0....0....1....0....1....0
%e A212408 ..1....0....1....1....1....0....1....0....1....1....1....1....0....0....0....0
%K A212408 nonn,new
%O A212408 1,1
%A A212408 _R. H. Hardin_ May 14 2012

%I A212407
%S A212407 34,166,747,3179,13185,54042,220054,892387,3609005,14567294,58714842,
%T A212407 236397086,950965002,3822869204,15359318444,61681353571,247609729669,
%U A212407 993662549686,3986465243314,15989373858826,64118439206974
%N A212407 Number of binary arrays of length 2*n+5 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212407 Row 6 of A212402
%H A212407 R. H. Hardin, <a href="/A212407/b212407.txt">Table of n, a(n) for n = 1..210</a>
%e A212407 Some solutions for n=3
%e A212407 ..1....1....1....0....0....0....1....1....1....0....0....0....0....0....1....1
%e A212407 ..0....1....0....1....0....1....1....1....0....0....1....0....1....1....1....0
%e A212407 ..1....0....1....0....0....1....0....0....0....0....0....1....1....0....0....0
%e A212407 ..0....1....0....1....1....0....1....1....0....1....0....0....0....0....1....0
%e A212407 ..0....0....0....1....0....0....0....0....1....0....1....0....0....1....0....0
%e A212407 ..1....0....0....0....0....0....0....0....1....0....0....0....0....0....0....1
%e A212407 ..1....1....0....0....1....0....0....1....0....0....0....1....1....1....0....0
%e A212407 ..0....0....0....1....0....0....1....1....0....1....1....1....0....1....0....0
%e A212407 ..0....0....1....0....0....0....1....0....0....1....0....0....0....0....0....1
%e A212407 ..0....1....0....0....1....1....1....0....0....0....1....1....0....0....1....0
%e A212407 ..1....0....1....0....0....1....0....1....1....1....1....0....0....1....1....1
%K A212407 nonn,new
%O A212407 1,1
%A A212407 _R. H. Hardin_ May 14 2012

%I A212406
%S A212406 21,97,421,1747,7143,29002,117290,473171,1905675,7665886,30810054,
%T A212406 123745422,496747206,1993227892,7995168852,32060722883,128532812627,
%U A212406 515187798518,2064622548782,8272744298618,33143688036722,132770436380108
%N A212406 Number of binary arrays of length 2*n+4 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212406 Row 5 of A212402
%H A212406 R. H. Hardin, <a href="/A212406/b212406.txt">Table of n, a(n) for n = 1..210</a>
%e A212406 Some solutions for n=3
%e A212406 ..0....1....1....0....0....0....1....0....0....0....1....0....1....0....0....1
%e A212406 ..1....0....0....0....1....0....1....0....1....1....1....0....1....1....1....0
%e A212406 ..1....1....0....0....0....0....1....0....1....0....0....1....0....0....0....0
%e A212406 ..0....0....0....1....0....0....0....1....0....1....0....0....0....0....0....1
%e A212406 ..0....1....0....0....0....0....0....0....1....0....0....1....1....1....0....1
%e A212406 ..0....0....1....0....0....0....0....0....0....0....0....1....0....0....1....0
%e A212406 ..0....1....0....0....0....1....1....1....0....0....0....0....0....1....0....0
%e A212406 ..1....0....0....0....1....0....1....1....1....1....1....0....0....0....1....1
%e A212406 ..0....0....0....0....0....0....0....0....1....1....0....0....0....0....0....0
%e A212406 ..1....1....1....1....1....1....1....1....0....1....1....0....1....0....0....1
%K A212406 nonn,new
%O A212406 1,1
%A A212406 _R. H. Hardin_ May 14 2012

%I A212405
%S A212405 13,57,236,959,3872,15586,62632,251419,1008536,4043582,16206152,
%T A212405 64933782,260114976,1041797124,4171943056,16704821779,66880877896,
%U A212405 267747443494,1071808583176,4290243456514,17172082337536,68729504287324
%N A212405 Number of binary arrays of length 2*n+3 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212405 Row 4 of A212402
%H A212405 R. H. Hardin, <a href="/A212405/b212405.txt">Table of n, a(n) for n = 1..210</a>
%e A212405 Some solutions for n=3
%e A212405 ..1....0....0....1....0....0....1....1....1....0....1....0....1....0....0....0
%e A212405 ..0....0....0....1....0....0....0....0....0....0....1....0....0....0....1....1
%e A212405 ..1....1....0....1....1....0....1....0....1....1....1....0....0....1....0....0
%e A212405 ..1....0....0....0....0....0....0....1....0....0....0....0....0....0....1....1
%e A212405 ..0....1....0....0....1....0....0....0....0....1....0....0....1....0....1....0
%e A212405 ..0....0....1....0....0....1....1....0....1....1....0....0....0....0....0....1
%e A212405 ..0....0....1....0....1....0....1....0....0....0....1....0....0....0....0....0
%e A212405 ..1....0....1....1....0....1....0....1....0....0....0....1....1....0....1....0
%e A212405 ..1....1....0....1....0....1....1....1....1....1....1....0....0....0....0....0
%K A212405 nonn,new
%O A212405 1,1
%A A212405 _R. H. Hardin_ May 14 2012

%I A212404
%S A212404 8,33,132,527,2104,8402,33560,134075,535728,2140910,8556568,34201078,
%T A212404 136713872,546528612,2184925808,8735357267,34925461088,139642914902,
%U A212404 558353310488,2232601256162,8927375430608,35698163696252,142750104755408
%N A212404 Number of binary arrays of length 2*n+2 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212404 Row 3 of A212402
%H A212404 R. H. Hardin, <a href="/A212404/b212404.txt">Table of n, a(n) for n = 1..210</a>
%e A212404 Some solutions for n=3
%e A212404 ..0....0....0....1....0....1....0....0....0....1....0....0....1....1....1....0
%e A212404 ..0....0....1....0....0....0....1....1....0....0....1....1....0....0....0....0
%e A212404 ..1....1....0....1....1....0....1....0....0....0....1....0....0....0....1....1
%e A212404 ..1....0....1....0....1....0....0....0....0....1....0....0....0....1....0....0
%e A212404 ..1....0....1....1....0....1....0....1....1....1....0....1....0....0....0....1
%e A212404 ..0....1....0....0....0....1....1....0....0....0....0....0....0....1....0....0
%e A212404 ..0....1....0....0....1....1....0....0....0....0....0....1....1....0....0....0
%e A212404 ..0....0....1....0....0....0....1....1....1....1....1....0....0....1....1....1
%K A212404 nonn,new
%O A212404 1,1
%A A212404 _R. H. Hardin_ May 14 2012

%I A212403
%S A212403 5,19,74,291,1150,4558,18100,71971,286454,1140954,4547020,18129294,
%T A212403 72309164,288493756,1151300584,4595507491,18346672294,73257044386,
%U A212403 292550538844,1168434892186,4667175448324,18644235526276,74485459541464
%N A212403 Number of binary arrays of length 2*n+1 with no more than n ones in any length 2n subsequence (=50% duty cycle)
%C A212403 Row 2 of A212402
%H A212403 R. H. Hardin, <a href="/A212403/b212403.txt">Table of n, a(n) for n = 1..210</a>
%e A212403 Some solutions for n=3
%e A212403 ..1....0....0....1....0....0....0....1....1....0....1....1....1....0....0....0
%e A212403 ..1....0....0....0....1....0....1....0....0....0....1....0....0....1....1....0
%e A212403 ..1....0....1....1....0....1....1....0....1....0....0....0....0....0....0....1
%e A212403 ..0....1....1....1....0....0....0....0....1....1....0....1....0....0....0....0
%e A212403 ..0....0....1....0....1....1....0....1....0....1....1....0....1....1....0....0
%e A212403 ..0....1....0....0....1....0....1....0....0....0....0....1....1....0....1....0
%e A212403 ..0....1....0....0....0....1....0....1....1....1....0....1....0....0....0....1
%K A212403 nonn,new
%O A212403 1,1
%A A212403 _R. H. Hardin_ May 14 2012

%I A212402
%S A212402 3,11,5,42,19,8,163,74,33,13,638,291,132,57,21,2510,1150,527,236,97,
%T A212402 34,9908,4558,2104,959,421,166,55,39203,18100,8402,3872,1747,747,285,
%U A212402 89,155382,71971,33560,15586,7143,3179,1314,489,144,616666,286454,134075
%N A212402 T(n,k)=Number of binary arrays of length n+2*k-1 with no more than k ones in any length 2k subsequence (=50% duty cycle)
%C A212402 Table starts
%C A212402 ..3..11...42...163...638...2510...9908...39203...155382...616666...2449868
%C A212402 ..5..19...74...291..1150...4558..18100...71971...286454..1140954...4547020
%C A212402 ..8..33..132...527..2104...8402..33560..134075...535728..2140910...8556568
%C A212402 .13..57..236...959..3872..15586..62632..251419..1008536..4043582..16206152
%C A212402 .21..97..421..1747..7143..29002.117290..473171..1905675..7665886..30810054
%C A212402 .34.166..747..3179.13185..54042.220054..892387..3609005.14567294..58714842
%C A212402 .55.285.1314..5769.24322.100736.413220.1685039..6844362.27724036.112072540
%C A212402 .89.489.2318.10425.44794.187696.776116.3183631.12990818.52815156.214150732
%H A212402 R. H. Hardin, <a href="/A212402/b212402.txt">Table of n, a(n) for n = 1..4360</a>
%e A212402 Some solutions for n=3 k=4
%e A212402 ..0....0....0....1....0....0....0....0....1....1....1....0....1....0....1....1
%e A212402 ..1....0....1....1....0....0....1....0....1....1....0....1....1....1....0....0
%e A212402 ..1....1....1....0....1....0....1....0....1....0....1....0....0....0....0....0
%e A212402 ..0....1....1....0....1....0....1....0....0....1....0....1....1....0....1....0
%e A212402 ..1....0....0....0....1....1....1....1....0....0....1....0....0....1....0....0
%e A212402 ..0....0....0....1....0....1....0....0....0....0....0....0....0....0....1....0
%e A212402 ..0....0....0....0....0....0....0....0....0....0....0....0....0....1....1....1
%e A212402 ..0....1....1....0....0....1....0....1....0....1....1....0....1....0....0....1
%e A212402 ..0....0....0....1....0....1....0....1....1....1....1....1....0....0....0....1
%e A212402 ..1....0....0....1....1....0....0....0....1....1....0....1....0....0....1....1
%Y A212402 Column 1 is A000045(n+3)
%Y A212402 Column 2 is A118647(n+3)
%Y A212402 Column 3 is A133551(n+5)
%Y A212402 Row 1 is A032443
%K A212402 nonn,tabl,new
%O A212402 1,1
%A A212402 _R. H. Hardin_ May 14 2012

%I A212401
%S A212401 9908,18100,33560,62632,117290,220054,413220,776116,1457281,2734307,
%T A212401 5124772,9591128,17917498,33399134,62096428,115525340,215173819,
%U A212401 401141813,748337734,1396704062,2607652294,4869423980,9093738783
%N A212401 Number of binary arrays of length n+13 with no more than 7 ones in any length 14 subsequence (=50% duty cycle)
%C A212401 Column 7 of A212402
%H A212401 R. H. Hardin, <a href="/A212401/b212401.txt">Table of n, a(n) for n = 1..210</a>
%e A212401 Some solutions for n=3
%e A212401 ..0....0....0....1....1....1....1....0....1....0....0....0....0....0....1....0
%e A212401 ..0....0....1....0....0....0....1....0....0....0....0....1....0....0....1....1
%e A212401 ..0....1....1....0....1....1....1....0....1....1....1....0....0....0....1....1
%e A212401 ..0....1....1....0....0....1....1....1....0....1....1....1....1....1....0....0
%e A212401 ..0....0....0....0....1....1....0....0....0....0....1....1....1....0....0....0
%e A212401 ..0....0....1....0....1....1....1....1....1....1....0....0....0....1....0....1
%e A212401 ..0....0....0....1....0....0....0....0....0....0....1....0....0....0....0....0
%e A212401 ..0....0....0....0....1....0....1....1....0....0....0....0....1....1....0....0
%e A212401 ..0....0....0....0....0....0....1....1....0....0....0....1....0....0....0....0
%e A212401 ..1....0....0....0....0....0....0....1....1....0....0....0....0....0....1....1
%e A212401 ..0....0....0....1....1....1....0....0....1....1....0....1....0....0....1....0
%e A212401 ..1....0....0....0....0....0....0....0....1....1....0....1....1....0....0....1
%e A212401 ..1....1....1....1....0....0....0....1....0....0....0....0....0....1....0....1
%e A212401 ..1....0....1....0....0....0....0....0....1....1....0....0....0....0....1....0
%e A212401 ..0....0....0....0....1....0....0....0....0....0....0....0....1....1....1....1
%e A212401 ..1....1....0....1....0....1....0....0....0....1....0....0....0....0....0....0
%K A212401 nonn,new
%O A212401 1,1
%A A212401 _R. H. Hardin_ May 14 2012

%I A212400
%S A212400 2510,4558,8402,15586,29002,54042,100736,187696,349360,649232,1203940,
%T A212400 2226644,4104676,7573112,13992148,25879040,47898464,88693997,
%U A212400 164277882,304304060,563672655,1043992249,1933271058,3579337110,6625751995
%N A212400 Number of binary arrays of length n+11 with no more than 6 ones in any length 12 subsequence (=50% duty cycle)
%C A212400 Column 6 of A212402
%H A212400 R. H. Hardin, <a href="/A212400/b212400.txt">Table of n, a(n) for n = 1..210</a>
%e A212400 Some solutions for n=3
%e A212400 ..0....1....1....1....1....1....0....0....0....1....0....1....0....0....1....1
%e A212400 ..0....1....1....1....1....1....0....1....1....0....0....1....0....0....1....1
%e A212400 ..1....0....0....0....0....0....1....0....0....0....1....0....1....0....0....0
%e A212400 ..1....0....0....0....0....1....1....0....0....1....0....0....0....0....1....0
%e A212400 ..0....1....0....0....1....0....1....0....0....0....0....1....1....1....1....0
%e A212400 ..0....0....1....1....0....1....0....1....0....1....1....0....0....0....0....1
%e A212400 ..1....0....0....0....0....0....1....1....0....0....0....0....0....0....1....1
%e A212400 ..1....0....0....0....0....1....0....1....1....1....1....1....0....1....1....1
%e A212400 ..0....0....0....0....0....0....0....0....0....1....0....0....0....0....0....0
%e A212400 ..0....0....1....0....0....1....0....0....1....0....0....0....0....1....0....1
%e A212400 ..0....1....1....0....0....0....0....1....0....1....1....1....0....1....0....0
%e A212400 ..0....1....0....0....1....0....1....0....1....0....0....1....0....0....0....0
%e A212400 ..0....1....1....0....1....0....0....1....1....0....0....0....1....1....0....1
%e A212400 ..1....0....0....1....1....0....0....0....0....1....1....1....1....0....1....0
%K A212400 nonn,new
%O A212400 1,1
%A A212400 _R. H. Hardin_ May 14 2012

%I A212399
%S A212399 638,1150,2104,3872,7143,13185,24322,44794,82294,150686,274744,501524,
%T A212399 917116,1679070,3076254,5638155,10334814,18942534,34712677,63595283,
%U A212399 116481088,213319856,390670798,715523414,1310607210,2400758159
%N A212399 Number of binary arrays of length n+9 with no more than 5 ones in any length 10 subsequence (=50% duty cycle)
%C A212399 Column 5 of A212402
%H A212399 R. H. Hardin, <a href="/A212399/b212399.txt">Table of n, a(n) for n = 1..210</a>
%e A212399 Some solutions for n=3
%e A212399 ..1....0....0....0....1....1....1....1....1....1....1....1....1....0....1....0
%e A212399 ..0....0....1....1....0....0....1....0....1....1....1....0....0....0....1....0
%e A212399 ..1....0....0....0....1....0....0....0....0....0....0....0....1....0....0....1
%e A212399 ..0....0....1....1....0....0....1....1....0....0....1....0....0....0....0....0
%e A212399 ..1....0....1....0....0....1....1....0....0....0....0....0....0....0....0....1
%e A212399 ..1....1....1....0....1....0....0....1....1....1....1....0....1....0....0....0
%e A212399 ..0....0....0....0....0....1....0....1....1....0....0....0....0....0....1....0
%e A212399 ..0....1....0....1....0....1....0....0....0....0....0....1....0....0....1....1
%e A212399 ..1....1....1....1....1....0....1....1....1....0....0....1....0....1....0....0
%e A212399 ..0....0....0....1....0....0....0....0....0....1....1....1....0....1....1....1
%e A212399 ..1....1....0....0....1....0....0....1....1....1....0....0....1....0....1....0
%e A212399 ..0....0....1....1....0....0....1....0....1....0....0....1....0....0....1....0
%K A212399 nonn,new
%O A212399 1,1
%A A212399 _R. H. Hardin_ May 14 2012

%I A212398
%S A212398 163,291,527,959,1747,3179,5769,10425,18729,33706,60797,109800,198415,
%T A212398 358592,647959,1170415,2113372,3815438,6888722,12439093,22463681,
%U A212398 40568913,73266801,132315810,238948339,431506179,779231793,1407175435
%N A212398 Number of binary arrays of length n+7 with no more than 4 ones in any length 8 subsequence (=50% duty cycle)
%C A212398 Column 4 of A212402
%H A212398 R. H. Hardin, <a href="/A212398/b212398.txt">Table of n, a(n) for n = 1..210</a>
%F A212398 Empirical: a(n) = a(n-1) +a(n-2) +a(n-4) -a(n-6) +3*a(n-7) +8*a(n-8) -8*a(n-10) -8*a(n-11) -10*a(n-12) -5*a(n-13) +8*a(n-14) -2*a(n-15) -28*a(n-16) -15*a(n-17) +25*a(n-18) +24*a(n-19) +28*a(n-20) +24*a(n-21) -19*a(n-22) -18*a(n-23) +51*a(n-24) +40*a(n-25) -55*a(n-26) -16*a(n-27) -55*a(n-28) -45*a(n-29) +51*a(n-30) +36*a(n-31) -61*a(n-32) -45*a(n-33) +70*a(n-34) -16*a(n-35) +67*a(n-36) +40*a(n-37) -70*a(n-38) -19*a(n-39) +56*a(n-40) +24*a(n-41) -58*a(n-42) +24*a(n-43) -56*a(n-44) -15*a(n-45) +56*a(n-46) -2*a(n-47) -28*a(n-48) -5*a(n-49) +28*a(n-50) -8*a(n-51) +28*a(n-52) -28*a(n-54) +3*a(n-55) +8*a(n-56) -8*a(n-58) -8*a(n-60) +a(n-61) +8*a(n-62) -a(n-64) +a(n-66) +a(n-68) -a(n-70)
%e A212398 Some solutions for n=3
%e A212398 ..0....0....1....0....0....0....1....0....0....0....1....1....0....1....1....0
%e A212398 ..1....0....0....0....0....0....0....0....0....1....0....0....0....1....0....0
%e A212398 ..1....0....1....0....1....1....0....0....0....1....1....1....0....0....0....0
%e A212398 ..0....1....0....1....0....0....0....0....0....0....0....0....0....1....0....0
%e A212398 ..0....0....0....0....0....1....1....1....0....1....1....1....0....0....0....0
%e A212398 ..0....0....1....0....1....0....1....0....1....0....0....0....0....0....1....1
%e A212398 ..0....1....1....1....1....0....0....0....1....1....0....1....0....1....0....0
%e A212398 ..0....0....0....0....0....0....0....1....1....0....1....0....0....0....1....0
%e A212398 ..1....1....0....0....0....1....1....0....0....0....1....0....0....1....1....1
%e A212398 ..0....0....1....0....0....1....1....0....1....1....0....1....0....1....1....1
%K A212398 nonn,new
%O A212398 1,1
%A A212398 _R. H. Hardin_ May 14 2012

%I A212326
%S A212326 1,4,20,112,676,4312,28704,197600,1397060,10090676,74152456,552666448,
%T A212326 4167528000,31736182776,243698432960,1884809367456,14668777816708,
%U A212326 114789815231560,902661488046900,7129068237647408,56524456978032904,449752267499647104
%N A212326 G.f. satisfies: A(x) = theta_3( x*A(x) )^2, where theta_3(x) is Jacobi's theta_3 function.
%F A212326 G.f. A(x) satisfies:
%F A212326 (1) A(x) = (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) )^2.
%F A212326 (2) A(x) =  1 + 4*Sum_{n>=1} (x*A(x))^n / (1 + (x*A(x))^(2*n)).
%F A212326 (3) A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n)^2 / (1 + (-x)^n*A(x)^n)^2.
%F A212326 (4) A( x/theta_3(x)^2 ) = theta_3(x)^2.
%F A212326 (5) A(x) = (1/x)*Series_Reversion(x/theta_3(x)^2), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%F A212326 a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1).
%e A212326 G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4312*x^5 + 28704*x^6 +...
%e A212326 Given g.f. A(x), let q = x*A(x), then by a q-series identity:
%e A212326 A(x) = 1 + 4*q/(1+q^2) + 4*q^2/(1+q^4) + 4*q^3/(1+q^6) + 4*q^4/(1+q^8) +...
%e A212326 A(x) = (1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...)^2.
%e A212326 ...
%e A212326 Illustrate a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1) by the following table of coefficients in powers theta_3(x)^(2*n+2) for n>=0:
%e A212326 n=0: [(1), 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0,...];
%e A212326 n=1: [1, (8), 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112,...];
%e A212326 n=2: [1, 12, (60), 160, 252, 312, 544, 960, 1020, 876, 1560, 2400,...];
%e A212326 n=3: [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, 12112,...];
%e A212326 n=4: [1, 20, 180, 960, (3380), 8424, 16320, 28800, 52020, 88660,...];
%e A212326 n=5: [1, 24, 264, 1760, 7944, (25872), 64416, 133056, 253704,...];
%e A212326 n=6: [1, 28, 364, 2912, 16044, 64792, (200928), 503360, ...];
%e A212326 n=7: [1, 32, 480, 4480, 29152, 140736, 525952, (1580800), ...]; ...
%e A212326 where the coefficients in parenthesis form the initial terms of this sequence:
%e A212326 A = [1/1, 8/2, 60/3, 448/4, 3380/5, 25872/6, 200928/7, 1580800/8, ...].
%o A212326 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+4*sum(m=1,n,(x*A)^m/(1+(x*A+x*O(x^n))^(2*m))));polcoeff(A,n)}
%o A212326 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=(1+2*sum(m=1,sqrtint(n+1),(x*A+x*O(x^n))^(m^2)))^2);polcoeff(A,n)}
%o A212326 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=prod(m=1,n,(1-(-x)^m*A^m)/(1+(-x)^m*A^m +x*O(x^n)))^2);polcoeff(A,n)}
%o A212326 for(n=0,30,print1(a(n),", "))
%Y A212326 Cf. A166952.
%K A212326 nonn,new
%O A212326 0,2
%A A212326 _Paul D. Hanna_, May 14 2012

%I A206495
%S A206495 2,3,4,5,6,7,6,6,8,10,11,13,17,9,10,12,14,13,13,14,19,12,12,12,16,15,
%T A206495 15,18,21,21,15,20,22,26,34,22,29,31,41,59,18,18,20,24,28,23,26,29,37,
%U A206495 43,21,26,26,28,38,25,30,33,35,39,51,24,24,24,24,32,34,41,41,43,67,27,30,30,36,42,42,37,37,37,38,53,30,30,40,44,52,68
%N A206495 Irregular triangle read by rows: row n contains in nondecreasing order the Matula-Goebel numbers of the elements of N(t), where t is the rooted tree with Matula-Goebel number n and N is the natural growth operator.
%C A206495 The natural growth operator maps a rooted tree t with V(t) vertices to the sequence of V(t) rooted trees, each having 1+V(t) vertices, by attaching one more outgoing edge and vertex to each vertex of t (the root remains the same). See, for example, the Brouder reference, p. 522 or the Connes-Kreimer reference, p. 225.
%C A206495 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 emenating 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.
%C A206495 Number of entries in row n is A061775(n).
%D A206495 A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys., 199, 203-242, 1998.
%D A206495 Ch. Brouder, Runge-Kutta methods and renormalization, Eur. Phys. J. C 12, 521-534, 2000.
%D A206495 F. Panaite, Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees, Letters Math. Phys., 51, 211-219, 2000.
%D A206495 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206495 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206495 I. Gutman and Y-N. Yeh, Deducing properties of trees from t heir Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206495 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A206495 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288v1 [math.CO].
%F A206495 Denote the natural growth operator by NG. It is convenient to introduce a "modified natural growth operator" MNG, defined just like NG, except that no edge is attached to the root. By NG(k) and MNG(k) we mean the action of these operators on the tree with Matula-Goebel number k. (i) NG(n) = [2n, MNG(n)]; (ii) MNG(1) = [ ]; (iii) if NG(t) = [a,b,c,...], then MNG(t-th prime) = [a-th prime, b-th prime, c-th prime, ...]; if r,s,>=2, then NG(rs) = [2rs,  r multiplied by the elements of MNG(s); s multiplied by the elements of MNG(r)]. The Maple program is based on these recurrence relations.
%e A206495 Row 2 is 3,4 because the rooted tree with Matula-Goebel number 2 is the 1-edge tree; attaching one edge at each vertex, we obtain \/ and the 2-edge path, having Matula-Goebel numbers 4 and 3, respectively.
%e A206495 Triangle starts:
%e A206495 2;
%e A206495 3,4;
%e A206495 5,6,7;
%e A206495 6,6,8;
%e A206495 10,11,13,17;
%e A206495 9,10,12,14;
%p A206495 with(numtheory): b := proc (n) local r, s, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: a := proc (n) options operator, arrow: [2*n, seq(b(n)[j], j = 1 .. nops(b(n)))] end proc: if n = 1 then [] elif bigomega(n) = 1 then map(ithprime, a(pi(n))) else [seq(r(n)*b(s(n))[j], j = 1 .. nops(b(s(n)))), seq(s(n)*b(r(n))[j], j = 1 .. nops(b(r(n))))] end if end proc: a := proc (n) options operator, arrow: sort([2*n, seq(b(n)[j], j = 1 .. nops(b(n)))]) end proc: for n to 20 do a(n) end do; # yields sequence in triangular form
%Y A206495 Cf. A061775.
%K A206495 nonn,tabf,new
%O A206495 1,1
%A A206495 _Emeric Deutsch_, May 13 2012

%I A212394
%S A212394 1,3,5,0,2,2,3,3,6,8,7,3,2,2,5,8,2,1,1,7,0,5,7,5,4,9,6,4,8,3,8,1,2,4,
%T A212394 7,1,0,3,6,0,4,2,6,1,3,8,8,9,3,5,3,6,3,3,4,8,4,9,3,7,2,7,5,7,0,9,9,5,
%U A212394 4,5,2,1,0,8,8,9,1,9,0,9,2,0,5,0,5,7,2,2,2,2,3,5,0,9,9,5,1,6,7,2
%N A212394 Decimal expansion of constant C = maximum value that sigma(n)*log(n^2)/n^2 reaches where sigma(n) = (sum of primes <= n), A034387.
%C A212394 From the prime number theorem it can be shown that the Prime sums function sigma(n) = (sum of primes <= n) ~ n^2/log(n^2). Consequently, the function sigma(n)*log(n^2)/n^2 tends to 1 as n tends to infinity, however it has a maximum value of 1.3502233687.... when n=7. In precise terms this constant is 34*log(7)/49 and it provides an upper bound for sigma(n), i.e. sigma(n) <= (34*log(7)/49)*n^2/log(n^2) for all n > 1.
%H A212394 J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>. Illinois J. Math. 6 (1962), pp. 64-94.
%F A212394 The maximum value for sigma(n)*log(n^2)/n^2 occurs at n = 7, so C = 34*log(7)/49.
%e A212394 1.350223368732258211705754964838124710360426138...
%t A212394 table=Table[Sum[Prime[k], {k, 1, PrimePi[n]}]/(n^2/(2 Log[n])), {n, 2, 10^4}]; max=Max[table]; n=1; While[table[[n]]!=max, n++]; Print[N[max, 100], " at n = ", n+1]
%o A212394 (PARI) log(7)*34/49 \\ _Charles R Greathouse IV_, May 14 2012
%Y A212394 Cf. A034387.
%K A212394 nonn,cons,new
%O A212394 1,2
%A A212394 _Frank M Jackson_, May 14 2012

%I A212390
%S A212390 1,1,1,1,1,1,1,1,1,1,1,2,13,79,365,1366,4369,12377,31825,75583,167961,
%T A212390 352718,705466,1352585,2501205,4495351,7956391,14221936,26802361,
%U A212390 56058016,133316626,350785307,967683665,2677259721,7246005881,18977267621,47931495649
%N A212390 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 10).
%C A212390 Lengths of descents are unrestricted.
%H A212390 Alois P. Heinz, <a href="/A212390/b212390.txt">Table of n, a(n) for n = 0..800</a>
%F A212390 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^10).
%e A212390 a(0) = 1: the empty path.
%e A212390 a(1) = 1: UD.
%e A212390 a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
%e A212390 a(12) = 13: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDDDDDDDDDDUDD, UUUUUUUUUUUDDDDDDDDDUDDD, UUUUUUUUUUUDDDDDDDDUDDDD, UUUUUUUUUUUDDDDDDDUDDDDD, UUUUUUUUUUUDDDDDDUDDDDDD, UUUUUUUUUUUDDDDDUDDDDDDD, UUUUUUUUUUUDDDDUDDDDDDDD, UUUUUUUUUUUDDDUDDDDDDDDD, UUUUUUUUUUUDDUDDDDDDDDDD, UUUUUUUUUUUDUDDDDDDDDDDD.
%p A212390 b:= proc(x, y, u) option remember;
%p A212390       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212390       `if`(u, add (b(x-(10*t+1), y, false), t=0..(x-1)/10), 0)))
%p A212390     end:
%p A212390 a:= n-> b(n$2, true):
%p A212390 seq (a(n), n=0..40);
%p A212390 # second Maple program
%p A212390 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^10), A), x, n+1), x, n):
%p A212390 seq (a(n), n=0..40);
%Y A212390 Column k=10 of A212382.
%K A212390 nonn,new
%O A212390 0,12
%A A212390 _Alois P. Heinz_, May 12 2012

%I A212389
%S A212389 1,1,1,1,1,1,1,1,1,1,2,12,67,287,1002,3004,8009,19449,43759,92380,
%T A212389 184787,353137,650497,1170632,2110021,3977161,8271836,19536661,
%U A212389 51111062,140210129,385123916,1032218316,2670065961,6645249777,15922990909,36823807747,82485177457
%N A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9).
%C A212389 Lengths of descents are unrestricted.
%H A212389 Alois P. Heinz, <a href="/A212389/b212389.txt">Table of n, a(n) for n = 0..800</a>
%F A212389 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).
%e A212389 a(0) = 1: the empty path.
%e A212389 a(1) = 1: UD.
%e A212389 a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
%e A212389 a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD.
%p A212389 b:= proc(x, y, u) option remember;
%p A212389       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212389       `if`(u, add (b(x-(9*t+1), y, false), t=0..(x-1)/9), 0)))
%p A212389     end:
%p A212389 a:= n-> b(n$2, true):
%p A212389 seq (a(n), n=0..40);
%p A212389 # second Maple program
%p A212389 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n):
%p A212389 seq (a(n), n=0..40);
%Y A212389 Column k=9 of A212382.
%K A212389 nonn,new
%O A212389 0,11
%A A212389 _Alois P. Heinz_, May 12 2012

%I A212388
%S A212388 1,1,1,1,1,1,1,1,1,2,11,56,221,716,2003,5006,11441,24312,48648,92721,
%T A212388 170811,311886,589590,1220979,2864973,7450852,20309628,55305706,
%U A212388 146505451,373452808,913836082,2150455648,4887179761,10794337952,23375638064,50219351232
%N A212388 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 8).
%C A212388 Lengths of descents are unrestricted.
%H A212388 Alois P. Heinz, <a href="/A212388/b212388.txt">Table of n, a(n) for n = 0..750</a>
%F A212388 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^8).
%e A212388 a(0) = 1: the empty path.
%e A212388 a(1) = 1: UD.
%e A212388 a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
%e A212388 a(10) = 11: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDDDDDDDDUDD, UUUUUUUUUDDDDDDDUDDD, UUUUUUUUUDDDDDDUDDDD, UUUUUUUUUDDDDDUDDDDD, UUUUUUUUUDDDDUDDDDDD, UUUUUUUUUDDDUDDDDDDD, UUUUUUUUUDDUDDDDDDDD, UUUUUUUUUDUDDDDDDDDD.
%p A212388 b:= proc(x, y, u) option remember;
%p A212388       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212388       `if`(u, add (b(x-(8*t+1), y, false), t=0..(x-1)/8), 0)))
%p A212388     end:
%p A212388 a:= n-> b(n$2, true):
%p A212388 seq (a(n), n=0..40);
%p A212388 # second Maple program
%p A212388 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^8), A), x, n+1), x, n):
%p A212388 seq (a(n), n=0..40);
%Y A212388 Column k=8 of A212382.
%K A212388 nonn,new
%O A212388 0,10
%A A212388 _Alois P. Heinz_, May 12 2012

%I A212387
%S A212387 1,1,1,1,1,1,1,1,2,10,46,166,496,1288,3004,6437,12895,24583,45799,
%T A212387 87211,180235,420547,1087220,2941931,7927664,20705636,51886966,
%U A212387 124660576,288445186,648173927,1431655546,3156274456,7062245781,16256654077,38704049941,94853117381
%N A212387 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 7).
%C A212387 Lengths of descents are unrestricted.
%H A212387 Alois P. Heinz, <a href="/A212387/b212387.txt">Table of n, a(n) for n = 0..750</a>
%F A212387 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^7).
%e A212387 a(0) = 1: the empty path.
%e A212387 a(1) = 1: UD.
%e A212387 a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
%e A212387 a(9) = 10: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDDDDDDDUDD, UUUUUUUUDDDDDDUDDD, UUUUUUUUDDDDDUDDDD, UUUUUUUUDDDDUDDDDD, UUUUUUUUDDDUDDDDDD, UUUUUUUUDDUDDDDDDD, UUUUUUUUDUDDDDDDDD.
%p A212387 b:= proc(x, y, u) option remember;
%p A212387       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212387       `if`(u, add (b(x-(7*t+1), y, false), t=0..(x-1)/7), 0)))
%p A212387     end:
%p A212387 a:= n-> b(n$2, true):
%p A212387 seq (a(n), n=0..40);
%p A212387 # second Maple program
%p A212387 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^7), A), x, n+1), x, n):
%p A212387 seq (a(n), n=0..40);
%Y A212387 Column k=7 of A212382.
%K A212387 nonn,new
%O A212387 0,9
%A A212387 _Alois P. Heinz_, May 12 2012

%I A212386
%S A212386 1,1,1,1,1,1,1,2,9,37,121,331,793,1718,3454,6646,12841,26589,61813,
%T A212386 158918,426401,1134431,2914055,7171539,16967745,39008002,88529366,
%U A212386 202057561,471422866,1133448790,2799775102,7026467132,17684574313,44192085565,109081884957
%N A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).
%C A212386 Lengths of descents are unrestricted.
%H A212386 Alois P. Heinz, <a href="/A212386/b212386.txt">Table of n, a(n) for n = 0..700</a>
%F A212386 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^6).
%e A212386 a(0) = 1: the empty path.
%e A212386 a(1) = 1: UD.
%e A212386 a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
%e A212386 a(8) = 9: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDDDDDDUDD, UUUUUUUDDDDDUDDD, UUUUUUUDDDDUDDDD, UUUUUUUDDDUDDDDD, UUUUUUUDDUDDDDDD, UUUUUUUDUDDDDDDD.
%p A212386 b:= proc(x, y, u) option remember;
%p A212386       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212386       `if`(u, add (b(x-(6*t+1), y, false), t=0..(x-1)/6), 0)))
%p A212386     end:
%p A212386 a:= n-> b(n$2, true):
%p A212386 seq (a(n), n=0..40);
%p A212386 # second Maple program
%p A212386 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^6), A), x, n+1), x, n):
%p A212386 seq (a(n), n=0..40);
%Y A212386 Column k=6 of A212382.
%K A212386 nonn,new
%O A212386 0,8
%A A212386 _Alois P. Heinz_, May 12 2012

%I A212385
%S A212385 1,1,1,1,1,1,2,8,29,85,211,464,943,1873,3914,9101,23298,61915,162283,
%T A212385 409888,996456,2360486,5555333,13244114,32357022,80958851,205389082,
%U A212385 522000262,1317987172,3297123652,8190326857,20302864970,50482613327,126318440989
%N A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).
%C A212385 Lengths of descents are unrestricted.
%H A212385 Alois P. Heinz, <a href="/A212385/b212385.txt">Table of n, a(n) for n = 0..700</a>
%F A212385 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5).
%e A212385 a(0) = 1: the empty path.
%e A212385 a(1) = 1: UD.
%e A212385 a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
%e A212385 a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.
%p A212385 b:= proc(x, y, u) option remember;
%p A212385       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212385       `if`(u, add (b(x-(5*t+1), y, false), t=0..(x-1)/5), 0)))
%p A212385     end:
%p A212385 a:= n-> b(n$2, true):
%p A212385 seq (a(n), n=0..40);
%p A212385 # second Maple program
%p A212385 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^5), A), x, n+1), x, n):
%p A212385 seq (a(n), n=0..40);
%Y A212385 Column k=5 of A212382.
%K A212385 nonn,new
%O A212385 0,7
%A A212385 _Alois P. Heinz_, May 12 2012

%I A212384
%S A212384 1,1,1,1,1,2,7,22,57,128,268,573,1343,3434,9038,23374,58649,144400,
%T A212384 355992,892336,2280020,5892301,15253305,39347067,101177783,260255812,
%U A212384 671941182,1743500452,4542147622,11858732144,30983904244,80982376879,211831943129,554905957520
%N A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).
%C A212384 Lengths of descents are unrestricted.
%H A212384 Alois P. Heinz, <a href="/A212384/b212384.txt">Table of n, a(n) for n = 0..700</a>
%F A212384 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^4).
%e A212384 a(0) = 1: the empty path.
%e A212384 a(1) = 1: UD.
%e A212384 a(5) = 2: UDUDUDUDUD, UUUUUDDDDD.
%e A212384 a(6) = 7: UDUDUDUDUDUD, UDUUUUUDDDDD, UUUUUDDDDDUD, UUUUUDDDDUDD, UUUUUDDDUDDD, UUUUUDDUDDDD, UUUUUDUDDDDD.
%p A212384 b:= proc(x, y, u) option remember;
%p A212384       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212384       `if`(u, add (b(x-(4*t+1), y, false), t=0..(x-1)/4), 0)))
%p A212384     end:
%p A212384 a:= n-> b(n$2, true):
%p A212384 seq (a(n), n=0..40);
%p A212384 # second Maple program
%p A212384 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^4), A), x, n+1), x, n):
%p A212384 seq (a(n), n=0..40);
%Y A212384 Column k=4 of A212382.
%K A212384 nonn,new
%O A212384 0,6
%A A212384 _Alois P. Heinz_, May 12 2012

%I A212383
%S A212383 1,1,1,1,2,6,16,37,83,199,512,1343,3488,9011,23488,62094,165738,
%T A212383 444160,1193146,3216436,8709766,23683846,64611879,176730460,484593740,
%U A212383 1332018207,3669981318,10133197561,28032766982,77688769031,215665451243,599644845226
%N A212383 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 3).
%C A212383 Lengths of descents are unrestricted.
%H A212383 Alois P. Heinz, <a href="/A212383/b212383.txt">Table of n, a(n) for n = 0..650</a>
%F A212383 G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^3).
%e A212383 a(0) = 1: the empty path.
%e A212383 a(1) = 1: UD.
%e A212383 a(2) = 1: UDUD.
%e A212383 a(3) = 1: UDUDUD.
%e A212383 a(4) = 2: UDUDUDUD, UUUUDDDD.
%e A212383 a(5) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.
%p A212383 b:= proc(x, y, u) option remember;
%p A212383       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A212383       `if`(u, add (b(x-(3*t+1), y, false), t=0..(x-1)/3), 0)))
%p A212383     end:
%p A212383 a:= n-> b(n$2, true):
%p A212383 seq (a(n), n=0..40);
%p A212383 # second Maple program
%p A212383 a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^3), A), x, n+1), x, n):
%p A212383 seq (a(n), n=0..40);
%Y A212383 Column k=3 of A212382.
%K A212383 nonn,new
%O A212383 0,5
%A A212383 _Alois P. Heinz_, May 12 2012

%I A212382
%S A212382 1,1,1,1,1,1,1,1,2,1,1,1,1,5,1,1,1,1,2,14,1,1,1,1,1,5,42,1,1,1,1,1,2,
%T A212382 12,132,1,1,1,1,1,1,6,30,429,1,1,1,1,1,1,2,16,79,1430,1,1,1,1,1,1,1,7,
%U A212382 37,213,4862,1,1,1,1,1,1,1,2,22,83,584,16796,1
%N A212382 Number A(n,k) of Dyck n-paths all of whose ascents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A212382 Lengths of descents are unrestricted.
%H A212382 Alois P. Heinz, <a href="/A212382/b212382.txt">Antidiagonals n = 0..140, flattened</a>
%F A212382 G.f. of column k>0 satisfies: A_k(x) = 1+x*A_k(x)/(1-(x*A_k(x))^k), g.f. of column k=0: A_0(x) = 1/(1-x).
%e A212382 A(0,k) = 1: the empty path.
%e A212382 A(3,0) = 1: UDUDUD.
%e A212382 A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
%e A212382 A(3,2) = 2: UDUDUD, UUUDDD.
%e A212382 A(5,3) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.
%e A212382 Square array A(n,k) begins:
%e A212382 1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212382 1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212382 1,   2,  1,  1,  1,  1,  1,  1, ...
%e A212382 1,   5,  2,  1,  1,  1,  1,  1, ...
%e A212382 1,  14,  5,  2,  1,  1,  1,  1, ...
%e A212382 1,  42, 12,  6,  2,  1,  1,  1, ...
%e A212382 1, 132, 30, 16,  7,  2,  1,  1, ...
%e A212382 1, 429, 79, 37, 22,  8,  2,  1, ...
%p A212382 b:= proc(x, y, k, u) option remember;
%p A212382       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, k, true)+
%p A212382       `if`(u, add (b(x-(k*t+1), y, k, false), t=0..(x-1)/k), 0)))
%p A212382     end:
%p A212382 A:= (n, k)-> `if`(k=0, 1, b(n, n, k, true)):
%p A212382 seq (seq (A(n, d-n), n=0..d), d=0..15);
%p A212382 # second Maple program
%p A212382 A:= (n, k)-> `if`(k=0, 1, coeff (series (RootOf
%p A212382               (A||k=1+x*A||k/(1-(x*A||k)^k), A||k), x, n+1), x, n)):
%p A212382 seq (seq (A(n, d-n), n=0..d), d=0..15);
%Y A212382 Columns k=0-10 give: A000012, A000108, A101785, A212383, A212384, A212385, A212386, A212387, A212388, A212389, A212390.
%K A212382 nonn,tabl,new
%O A212382 0,9
%A A212382 _Alois P. Heinz_, May 12 2012

%I A212277
%S A212277 1,1,4,24,168,1284,10384,87364,756808,6704968,60471040,553334434,
%T A212277 5124366956,47938322744,452349133904,4300336433872,41148686798000,
%U A212277 396000558255084,3830370110005728,37218151946806512,363109794135657408,3555651588908143457,34934228253014629644
%N A212277  G.f. satisfies: A(x) = x + A(A(x)^2)^2 where g.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2).
%F A212277  Self-convolution yields A212392.
%e A212277  G.f.: A(x) = x + x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 +...
%e A212277 such that
%e A212277 A(A(x)^2)^2 = x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 +...
%e A212277 where
%e A212277 A(x)^2 = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...+ A212392(n)*x^(3*n-1) +...
%o A212277  (PARI) {a(n)=local(A=x+x^4); for(i=1, n, A=x+subst(A^2, x, A^2+O(x^(3*n)))); polcoeff(A, 3*n-2)}
%o A212277 for(n=1, 30, print1(a(n), ", "))
%Y A212277  Cf. A212392, A212391.
%K A212277 nonn,new
%O A212277 1,3
%A A212277 _Paul D. Hanna_, May 13 2012

%I A211921
%S A211921 0,1,11,58,176,407,840,1536,2591,4133,6268,9119,12895,17684,23706,
%T A211921 31201,40315,51239,64352,79770,97829,118795,142947,170548,202114,
%U A211921 237757,277915,323080,373489,429449,491670,560274,635851,718858,809692
%N A211921 Number of (w,x,y,z) with all terms in {1,...,n} and 2w*x<=3*y*z.
%C A211921 A211921(n)+A211922(n)=n^4.
%C A211921 See A211795 for a guide to related sequences.
%t A211921 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211921 (Do[If[2 w*x <= 3 y*z, s = s + 1],
%t A211921 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211921 Map[t[#] &, Range[0, 40]] (* A211921 *)
%t A211921 (* Peter Moses, Apr 13 2012 *)
%Y A211921 Cf. A211795.
%K A211921 nonn,new
%O A211921 0,3
%A A211921 _Clark Kimberling_, Apr 28 2012

%I A211922
%S A211922 0,0,5,23,80,218,456,865,1505,2428,3732,5522,7841,10877,14710,19424,
%T A211922 25221,32282,40624,50551,62171,75686,91309,109293,129662,152868,
%U A211922 179061,208361,241167,277832,318330,363247,412725,467063,526644,591736,662511,739687,823382
%N A211922 Number of (w,x,y,z) with all terms in {1,...,n} and 2*w*x>3*y*z.
%C A211922 A211922(n)+A211921(n)=n^4.
%C A211922 See A211795 for a guide to related sequences.
%t A211922 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211922 (Do[If[2 w*x > 3 y*z, s = s + 1],
%t A211922 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211922 Map[t[#] &, Range[0, 40]] (* A211922 *)
%t A211922 (* Peter Moses, Apr 13 2012 *)
%Y A211922 Cf. A211795.
%K A211922 nonn,new
%O A211922 0,3
%A A211922 _Clark Kimberling_, Apr 28 2012

%I A212019
%S A212019 0,0,4,8,24,32,64,76,124,154,214,234,340,364,452,532,664,696,854,890,
%T A212019 1086,1206,1350,1394,1692,1794,1966,2108,2396,2452,2822,2882,3212,
%U A212019 3408,3636,3860,4360,4432,4688,4920,5470,5550,6090,6174,6642,7056
%N A212019 Number of (w,x,y,z) with all terms in {1,...,n} and w*x=2*y*z.
%C A212019 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%e A212019 a(2) counts these four 4-tuples: (1,2,1,1), (2,1,1,1), (2,2,1,2), (2,2,2,1).
%t A212019 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212019 (Do[If[w*x == 2 y*z, s = s + 1],
%t A212019 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212019 Map[t[#] &, Range[0, 60]] (* A212019 *)
%t A212019 %/2  (* integers *)
%t A212019 (* Peter Moses, Apr 13 2012 *)
%Y A212019 Cf. A211795.
%K A212019 nonn,new
%O A212019 1,3
%A A212019 _Clark Kimberling_, Apr 28 2012

%I A212048
%S A212048 0,0,0,2,14,18,30,34,74,104,128,136,216,228,260,318,438,454,554,570,
%T A212048 718,812,868,888,1128,1182,1250,1398,1614,1642,1864,1892,2204,2354,
%U A212048 2446,2570,2990,3026,3126,3304,3760,3800,4128,4168,4520,4906,5030
%N A212048 Number of (w,x,y,z) with all terms in {1,...,n} and 3w*x=4*y*z.
%C A212048 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212048 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212048  (Do[If[3 w*x == 4 y*z, s = s + 1],
%t A212048  {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212048 Map[t[#] &, Range[0, 50]] (* A212048 *)
%t A212048 %/2  (* integers *)
%t A212048 (* Peter Moses, Apr 13 2012 *)
%Y A212048 Cf. A211795.
%K A212048 nonn,new
%O A212048 0,4
%A A212048 _Clark Kimberling_, Apr 29 2012

%I A212050
%S A212050 0,0,0,0,0,20,32,36,46,54,118,126,162,170,194,316,358,370,430,442,614,
%T A212050 666,714,730,840,1064,1124,1184,1276,1296,1686,1710,1824,1908,1980,
%U A212050 2320,2524,2552,2636,2732,3202,3234,3470,3502,3658,4222,4330
%N A212050 Number of (w,x,y,z) with all terms in {1,...,n} and 2w*x=5*y*z.
%C A212050 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212050 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212050 (Do[If[2 w*x == 5 y*z, s = s + 1],
%t A212050 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212050 Map[t[#] &, Range[0, 50]] (* A212050 *)
%t A212050 %/2  (* integers *)
%t A212050 (* Peter Moses, Apr 13 2012 *)
%Y A212050 Cf. A211795.
%K A212050 nonn,new
%O A212050 0,6
%A A212050 _Clark Kimberling_, Apr 29 2012

%I A212049
%S A212049 0,0,0,0,0,18,26,30,38,44,108,116,144,152,172,280,314,326,372,384,564,
%T A212049 612,652,668,752,954,1002,1056,1136,1156,1528,1552,1642,1718,1778,
%U A212049 2084,2250,2278,2346,2430,2922,2954,3150,3182,3318,3830,3918,3954
%N A212049 Number of (w,x,y,z) with all terms in {1,...,n} and w*x=5*y*z.
%C A212049 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212049 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212049 (Do[If[w*x == 5 y*z, s = s + 1],
%t A212049 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212049 Map[t[#] &, Range[0, 50]] (* A212049 *)
%t A212049 %/2  (* integers *)
%t A212049 (* Peter Moses, Apr 13 2012 *)
%Y A212049 Cf. A211795.
%K A212049 nonn,new
%O A212049 0,6
%A A212049 _Clark Kimberling_, Apr 29 2012

%I A212051
%S A212051 0,0,0,0,0,16,28,32,40,52,112,120,164,172,192,280,310,322,398,410,580,
%T A212051 644,680,696,816,1004,1048,1138,1218,1238,1592,1616,1702,1798,1854,
%U A212051 2138,2394,2422,2486,2598,3068,3100,3368,3400,3536,3976,4056
%N A212051 Number of (w,x,y,z) with all terms in {1,...,n} and 3w*x=5*y*z.
%C A212051 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212051 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212051 (Do[If[3 w*x == 5 y*z, s = s + 1],
%t A212051 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212051 Map[t[#] &, Range[0, 50]] (* A212051 *)
%t A212051 %/2  (* integers *)
%t A212051 (* Peter Moses, Apr 13 2012 *)
%Y A212051 Cf. A211795.
%K A212051 nonn,new
%O A212051 0,6
%A A212051 _Clark Kimberling_, Apr 29 2012

%I A212052
%S A212052 0,0,0,0,0,18,26,30,46,52,104,112,148,156,176,278,342,354,406,418,562,
%T A212052 602,642,658,794,988,1040,1090,1190,1210,1526,1550,1726,1794,1858,
%U A212052 2152,2368,2396,2468,2548,2980,3012,3212,3244,3408,3890,3982,4018
%N A212052 Number of (w,x,y,z) with all terms in {1,...,n} and 4w*x=5*y*z.
%C A212052 See A211795 for a guide to related sequences.
%t A212052 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212052 (Do[If[4 w*x == 5 y*z, s = s + 1],
%t A212052 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212052 Map[t[#] &, Range[0, 50]] (* A212052 *)
%t A212052 %/2  (* integers *)
%t A212052 (* Peter Moses, Apr 13 2012 *)
%Y A212052 Cf. A211795.
%K A212052 nonn,new
%O A212052 0,6
%A A212052 _Clark Kimberling_, Apr 29 2012

%I A212021
%S A212021 0,0,0,8,16,20,48,56,80,132,164,176,272,288,332,436,512,532,696,720,
%T A212021 844,1000,1076,1104,1360,1430,1518,1758,1942,1978,2328,2368,2566,2818,
%U A212021 2938,3098,3600,3648,3780,4084,4440,4492,4996,5052,5348,5920
%N A212021 Number of (w,x,y,z) with all terms in {1,...,n} and 2w*x=3*y*z.
%C A212021 Each term is divisible by 2.  See A211795 for a guide to related sequences.
%t A212021 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212021 (Do[If[2 w*x == 3 y*z, s = s + 1],
%t A212021 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212021 Map[t[#] &, Range[0, 50]] (* A212021 *)
%t A212021 %/2  (* integers *)
%t A212021 (* Peter Moses, Apr 13 2012 *)
%Y A212021 Cf. A211795.
%K A212021 nonn,new
%O A212021 0,4
%A A212021 _Clark Kimberling_, Apr 28 2012

%I A211920
%S A211920 0,1,11,50,160,387,792,1480,2511,4001,6104,8943,12623,17396,23374,
%T A211920 30765,39803,50707,63656,79050,96985,117795,141871,169444,200754,
%U A211920 236327,276397,321322,371547,427471,489342,557906,633285,716040,806754
%N A211920 Number of (w,x,y,z) with all terms in {1,...,n} and 2*w*x<3*y*z.
%C A211920 A211920(n)+A211923(n)=n^4.
%C A211920 See A211795 for a guide to related sequences.
%t A211920 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211920 (Do[If[2 w*x < 3 y*z, s = s + 1],
%t A211920 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211920 Map[t[#] &, Range[0, 40]] (* A211920 *)
%t A211920 (* Peter Moses, Apr 13 2012 *)
%Y A211920 Cf. A211795.
%K A211920 nonn,new
%O A211920 0,3
%A A211920 _Clark Kimberling_, Apr 28 2012

%I A212391
%S A212391 1,1,3,14,80,516,3608,26729,206808,1655232,13612512,114466491,
%T A212391 980575020,8533242324,75267759072,671721353474,6056517394512,
%U A212391 55104831724236,505422858053560,4669306663437888,43418090784597696,406109012334694211,3818890067546807794
%N A212391 a(n) = A212392(n) / n.
%H A212391 Paul D. Hanna, <a href="/A212391/b212391.txt">Table of n, a(n) for n = 1..200</a>
%F A212391 Given g.f. A(x), then G(x) = d/dx A(x^3)/3 = Sum_{n>=1} n*a(n)*x^(3*n-1) is the g.f. of A212392 and satisfies: G(x) = (x + G(G(x)))^2.
%F A212391 G.f. satisfies: A’(x) = ( 1 + x*A’(x)^2 * A’(x^2*A’(x)^3) )^2 where A'(x) = d/dx A(x).
%e A212391 G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 80*x^5 + 516*x^6 + 3608*x^7 + 26729*x^8 +...
%e A212391 Let G(x) = d/dx A(x^3)/3, then G(x) = (x + G(G(x)))^2, where
%e A212391 G(x) = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...
%e A212391 G(G(x)) = x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 +...
%o A212391 (PARI) {a(n)=local(G=x^2+x^3);for(i=1,n,G=(x+subst(G,x,G+O(x^(3*n))))^2);polcoeff(G,3*n-1)/n}
%o A212391 for(n=1,30,print1(a(n),", "))
%Y A212391 Cf. A212392.
%K A212391 nonn,new
%O A212391 1,3
%A A212391 _Paul D. Hanna_, May 12 2012

%I A212392
%S A212392 1,2,9,56,400,3096,25256,213832,1861272,16552320,149737632,1373597892,
%T A212392 12747475260,119465392536,1129016386080,10747541655584,
%U A212392 102960795706704,991886971036248,9603034303017640,93386133268757760,911779906476551616,8934398271363272642
%N A212392 G.f. satisfies: A(x) = (x + A(A(x)))^2 where g.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-1).
%C A212392 Conjecture: n divides a(n); see A212391.
%C A212392 More generally, we have the conjecture:
%C A212392 If    A(x) = ( x + A(A(x)) )^b
%C A212392 where A(x) = Sum_{n>=1} a(n) * x^((b^2-1)*(n-1)+b)
%C A212392 then  ((b-1)*(n-1)+1) divides a(n).
%H A212392 Paul D. Hanna, <a href="/A212392/b212392.txt">Table of n, a(n) for n = 1..200</a>
%F A212392 a(n) = n*A212391(n).
%e A212392 G.f.: A(x) = x^2 + 2*x^5 + 9*x^8 + 56*x^11 + 400*x^14 + 3096*x^17 + 25256*x^20 +...
%e A212392 such that A(x) = (x + A(A(x)))^2, where
%e A212392 A(A(x)) = x^4 + 4*x^7 + 24*x^10 + 168*x^13 + 1284*x^16 + 10384*x^19 + 87364*x^22 + 756808*x^25 + 6704968*x^28 + 60471040*x^31 +...+ A212277(n+1)*x^(3*n+1) +...
%e A212392 Note that sqrt(A(A(x))) = A(x) + A(A(A(x))), where
%e A212392 sqrt(A(A(x))) = x^2 + 2*x^5 + 10*x^8 + 64*x^11 + 464*x^14 + 3624*x^17 + 29746*x^20 + 252976*x^23 + 2209488*x^26 + 19701504*x^29 +...
%e A212392 A(A(A(x))) = x^8 + 8*x^11 + 64*x^14 + 528*x^17 + 4490*x^20 + 39144*x^23 + 348216*x^26 + 3149184*x^29 + 28872401*x^32 +...
%o A212392 (PARI) {a(n)=local(A=x^2+x^3);for(i=1,n,A=(x+subst(A,x,A+O(x^(3*n))))^2);polcoeff(A,3*n-1)}
%o A212392 for(n=1,30,print1(a(n),", "))
%o A212392 (Maxima) A(n,k):= if  n<2*k then 0 else if n/2=k then 1 else sum(binomial(2*k,j)*sum(A(i,2*k-j)*A(n-j,i),i,2*k-j+1,n-j-1),j,0,2*k-1);
%o A212392 makelist(A(n,1),n,1,17); [_Vladimir Kruchinin_, May 15 2012]
%Y A212392 Cf. A212391, A212277.
%K A212392 nonn,new
%O A212392 1,2
%A A212392 _Paul D. Hanna_, May 12 2012

%I A212047
%S A212047 0,0,1,1,14,18,31,35,80,96,125,133,216,228,269,309,444,460,543,559,
%T A212047 720,784,853,873,1142,1202,1287,1363,1596,1624,1831,1859,2206,2314,
%U A212047 2427,2559,2962,2998,3123,3247,3758,3798,4103,4143,4524,4772,4925,4969,5712,5836
%N A212047 Number of (w,x,y,z) with all terms in {1,...,n} and w*x=4*y*z.
%C A212047 See A211795 for a guide to related sequences.
%t A212047 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212047 (Do[If[w*x == 4 y*z, s = s + 1],
%t A212047 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212047 Map[t[#] &, Range[0, 50]] (* A212047 *)
%t A212047 (* Peter Moses, Apr 13 2012 *)
%Y A212047 Cf. A211795.
%K A212047 nonn,new
%O A212047 0,5
%A A212047 _Clark Kimberling_, Apr 28 2012

%I A212086
%S A212086 0,0,3,10,25,49,86,137,206,294,405,540,703,895,1120,1379,1676,2012,
%T A212086 2391,2814,3285,3805,4378,5005,5690,6434,7241,8112,9051,10059,11140,
%U A212086 12295,13528,14840,16235,17714,19281,20937,22686,24529,26470,28510
%N A212086 Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.
%C A212086 For a guide to related sequences, see A211795.
%F A212086 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5).
%F A212086 G.f.: x^2*(x^2+x+3)/((x+1)*(-1+x)^4).
%t A212086 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212086 (Do[If[2 w == x + y + z, s = s + 1],
%t A212086 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212086 Map[t[#] &, Range[0, 50]] (* A212086 *)
%t A212086 FindLinearRecurrence[%]
%t A212086 (* Peter Moses, Apr 13 2012 *)
%Y A212086 Cf. A211795.
%K A212086 nonn,new
%O A212086 0,3
%A A212086 _Clark Kimberling_, May 01 2012

%I A211923
%S A211923 0,0,5,31,96,238,504,921,1585,2560,3896,5698,8113,11165,15042,19860,
%T A211923 25733,32814,41320,51271,63015,76686,92385,110397,131022,154298,
%U A211923 180579,210119,243109,279810,320658,365615,415291,469881,529582,594834,666111,743335,827162
%N A211923 Number of (w,x,y,z) with all terms in {1,...,n} and 2*w*x>=3*y*z.
%C A211923 A211923(n)+A211920(n)=n^4.
%C A211923 See A211795 for a guide to related sequences.
%t A211923 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211923 (Do[If[2 w*x >= 3 y*z, s = s + 1],
%t A211923 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211923 Map[t[#] &, Range[0, 40]] (* A211923 *)
%t A211923 (* Peter Moses, Apr 13 2012 *)
%Y A211923 Cf. A211795.
%K A211923 nonn,new
%O A211923 0,3
%A A211923 _Clark Kimberling_, Apr 28 2012

%I A210935
%S A210935 1,4,64,80,108,270,351,432,729,768,864,2916,5184,5832,6250,6912,12096,
%T A210935 13500,16384,25600,32832,34992,37500,39366,43200,46656,50000,73008,
%U A210935 74304,81648,84375,110592,131250,138240,143748,153664,172800,176418,200000,225000
%N A210935 Numbers n for which n*n'/(n+n') is an integer, where n' is the arithmetic derivative of n.
%C A210935 Only eleven odd numbers in the first 150 terms: a(1)=1, a(7)=351, a(9)=729, a(31)=84375, a(76)=2470629,  a(78)=2709375, a(87)=4159375, a(89)=4348377, a(115)=13286025, a(126)=22235661, a(128)=25059375.
%H A210935 Paolo P. Lava, <a href="/A210935/b210935.txt">Table of n, a(n) for n = 1..150</a>
%e A210935 n=729; n'=1458; n*n'/(n+n')=486.
%p A210935 with(numtheory);
%p A210935 A210935:= proc(n)
%p A210935 local a,i,p,pfs;
%p A210935 for i from 1 to n do
%p A210935     pfs:=ifactors(i)[2];  a:=i*add(op(2,p)/op(1,p),p=pfs) ;
%p A210935     if a*i/(a+i)=trunc(a*i/(a+i)) then print(i); fi;
%p A210935 od; end:
%p A210935 A210935(100000000);
%Y A210935 Cf. A003415, A165562.
%K A210935 nonn,new
%O A210935 1,2
%A A210935 _Paolo P. Lava_, May 11 2012

%I A212369
%S A212369 1,1,1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,46,56,68,85,112,156,226,
%T A212369 333,490,712,1016,1421,1949,2630,3512,4676,6256,8464,11620,16187,
%U A212369 22811,32366,46005,65225,91967,128786,179140,247861,341885,471332,651041,902679
%N A212369 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 10).
%H A212369 Alois P. Heinz, <a href="/A212369/b212369.txt">Table of n, a(n) for n = 0..1000</a>
%F A212369 G.f. satisfies: A(x) = 1+A(x)*(x-x^10*(1-A(x))).
%F A212369 a(n) = a(n-1) + Sum_{k=1..n-10} a(k)*a(n-10-k) if n>0; a(0) = 1.
%e A212369 a(0) = 1: the empty path.
%e A212369 a(1) = 1: UD.
%e A212369 a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
%e A212369 a(12) = 4: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDUDDDDDDDDDDD.
%p A212369 a:= proc(n) option remember;
%p A212369       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-10-k), k=1..n-10))
%p A212369     end:
%p A212369 seq (a(n), n=0..60);
%p A212369 # second Maple program
%p A212369 a:= n-> coeff(series(RootOf(A=1+A*(x-x^10*(1-A)), A), x, n+1), x, n):
%p A212369 seq (a(n), n=0..60);
%Y A212369 Column k=10 of A212363.
%K A212369 nonn,new
%O A212369 0,12
%A A212369 _Alois P. Heinz_, May 10 2012

%I A212368
%S A212368 1,1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,46,57,73,99,142,211,317,473,
%T A212368 694,997,1402,1937,2648,3614,4967,6917,9782,14023,20284,29438,42647,
%U A212368 61457,87963,125093,177074,250157,353692,501658,714768,1023296,1470843
%N A212368 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 9).
%H A212368 Alois P. Heinz, <a href="/A212368/b212368.txt">Table of n, a(n) for n = 0..1000</a>
%F A212368 G.f. satisfies: A(x) = 1+A(x)*(x-x^9*(1-A(x))).
%F A212368 a(n) = a(n-1) + Sum_{k=1..n-9} a(k)*a(n-9-k) if n>0; a(0) = 1.
%e A212368 a(0) = 1: the empty path.
%e A212368 a(1) = 1: UD.
%e A212368 a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
%e A212368 a(11) = 4: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDUDDDDDDDDDD.
%p A212368 a:= proc(n) option remember;
%p A212368       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-9-k), k=1..n-9))
%p A212368     end:
%p A212368 seq (a(n), n=0..60);
%p A212368 # second Maple program
%p A212368 a:= n-> coeff(series(RootOf(A=1+A*(x-x^9*(1-A)), A), x, n+1), x, n):
%p A212368 seq (a(n), n=0..60);
%Y A212368 Column k=9 of A212363.
%K A212368 nonn,new
%O A212368 0,11
%A A212368 _Alois P. Heinz_, May 10 2012

%I A212367
%S A212367 1,1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,37,47,62,87,129,197,302,457,677,
%T A212367 980,1392,1957,2752,3907,5630,8237,12187,18123,26927,39810,58472,
%U A212367 85381,124234,180677,263375,385538,567036,837306,1239408,1835867,2717386,4016173
%N A212367 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 8).
%H A212367 Alois P. Heinz, <a href="/A212367/b212367.txt">Table of n, a(n) for n = 0..1000</a>
%F A212367 G.f. satisfies: A(x) = 1+A(x)*(x-x^8*(1-A(x))).
%F A212367 a(n) = a(n-1) + Sum_{k=1..n-8} a(k)*a(n-8-k) if n>0; a(0) = 1.
%e A212367 a(0) = 1: the empty path.
%e A212367 a(1) = 1: UD.
%e A212367 a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
%e A212367 a(10) = 4: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDUDDDDDDDDD.
%p A212367 a:= proc(n) option remember;
%p A212367       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-8-k), k=1..n-8))
%p A212367     end:
%p A212367 seq (a(n), n=0..60);
%p A212367 # second Maple program
%p A212367 a:= n-> coeff(series(RootOf(A=1+A*(x-x^8*(1-A)), A), x, n+1), x, n):
%p A212367 seq (a(n), n=0..60);
%Y A212367 Column k=8 of A212363.
%K A212367 nonn,new
%O A212367 0,10
%A A212367 _Alois P. Heinz_, May 10 2012

%I A212366
%S A212366 1,1,1,1,1,1,1,1,2,4,7,11,16,22,29,38,52,76,117,184,288,442,662,972,
%T A212366 1414,2063,3047,4572,6952,10645,16303,24857,37672,56821,85541,128948,
%U A212366 195103,296548,452501,692053,1058990,1619311,2473171,3773889,5757885,8791090
%N A212366 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7).
%H A212366 Alois P. Heinz, <a href="/A212366/b212366.txt">Table of n, a(n) for n = 0..1000</a>
%F A212366 G.f. satisfies: A(x) = 1+A(x)*(x-x^7*(1-A(x))).
%F A212366 a(n) = a(n-1) + Sum_{k=1..n-7} a(k)*a(n-7-k) if n>0; a(0) = 1.
%e A212366 a(0) = 1: the empty path.
%e A212366 a(1) = 1: UD.
%e A212366 a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
%e A212366 a(9) = 4: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDUDDDDDDDD.
%p A212366 a:= proc(n) option remember;
%p A212366       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-7-k), k=1..n-7))
%p A212366     end:
%p A212366 seq (a(n), n=0..50);
%p A212366 # second Maple program
%p A212366 a:= n-> coeff(series(RootOf(A=1+A*(x-x^7*(1-A)), A), x, n+1), x, n):
%p A212366 seq (a(n), n=0..50);
%Y A212366 Column k=7 of A212363.
%K A212366 nonn,new
%O A212366 0,9
%A A212366 _Alois P. Heinz_, May 10 2012

%I A212365
%S A212365 1,1,1,1,1,1,1,2,4,7,11,16,22,30,43,66,106,172,275,429,656,996,1522,
%T A212365 2360,3714,5897,9376,14852,23410,36788,57828,91187,144413,229561,
%U A212365 365678,582766,928280,1477877,2353062,3749721,5983631,9562565,15300700,24501417
%N A212365 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 6).
%H A212365 Alois P. Heinz, <a href="/A212365/b212365.txt">Table of n, a(n) for n = 0..1000</a>
%F A212365 G.f. satisfies: A(x) = 1+A(x)*(x-x^6*(1-A(x))).
%F A212365 a(n) = a(n-1) + Sum_{k=1..n-6} a(k)*a(n-6-k) if n>0; a(0) = 1.
%e A212365 a(0) = 1: the empty path.
%e A212365 a(1) = 1: UD.
%e A212365 a(6) = 1: UDUDUDUDUDUD.
%e A212365 a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
%e A212365 a(8) = 4: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDUDDDDDDD.
%p A212365 a:= proc(n) option remember;
%p A212365       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-6-k), k=1..n-6))
%p A212365     end:
%p A212365 seq (a(n), n=0..50);
%p A212365 # second Maple program
%p A212365 a:= n-> coeff(series(RootOf(A=1+A*(x-x^6*(1-A)), A), x, n+1), x, n):
%p A212365 seq (a(n), n=0..50);
%Y A212365 Column k=6 of A212363.
%K A212365 nonn,new
%O A212365 0,8
%A A212365 _Alois P. Heinz_, May 10 2012

%I A212364
%S A212364 1,1,1,1,1,1,2,4,7,11,16,23,35,57,96,161,264,425,682,1106,1821,3030,
%T A212364 5055,8412,13956,23145,38487,64261,107673,180762,303651,510187,857692,
%U A212364 1443597,2433495,4108299,6943862,11746362,19883655,33681015,57096874,96874214
%N A212364 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 5).
%H A212364 Alois P. Heinz, <a href="/A212364/b212364.txt">Table of n, a(n) for n = 0..1000</a>
%F A212364 G.f. satisfies: A(x) = 1+A(x)*(x-x^5*(1-A(x))).
%F A212364 a(n) = a(n-1) + Sum_{k=1..n-5} a(k)*a(n-5-k) if n>0; a(0) = 1.
%e A212364 a(0) = 1: the empty path.
%e A212364 a(1) = 1: UD.
%e A212364 a(5) = 1: UDUDUDUDUD.
%e A212364 a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
%e A212364 a(7) = 4: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDUDDDDDD.
%e A212364 a(8) = 7: UDUDUDUDUDUDUDUD, UDUDUUUUUUDDDDDD, UDUUUUUUDDDDDDUD, UDUUUUUUDUDDDDDD, UUUUUUDDDDDDUDUD, UUUUUUDUDDDDDDUD, UUUUUUDUDUDDDDDD.
%p A212364 a:= proc(n) option remember;
%p A212364       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-5-k), k=1..n-5))
%p A212364     end:
%p A212364 seq (a(n), n=0..50);
%p A212364 # second Maple program
%p A212364 a:= n-> coeff(series(RootOf(A=1+A*(x-x^5*(1-A)), A), x, n+1), x, n):
%p A212364 seq (a(n), n=0..50);
%Y A212364 Column k=5 of A212363.
%K A212364 nonn,new
%O A212364 0,7
%A A212364 _Alois P. Heinz_, May 10 2012

%I A212363
%S A212363 1,1,1,1,1,1,1,1,2,1,1,1,1,5,1,1,1,1,2,14,1,1,1,1,1,4,42,1,1,1,1,1,2,
%T A212363 8,132,1,1,1,1,1,1,4,17,429,1,1,1,1,1,1,2,7,37,1430,1,1,1,1,1,1,1,4,
%U A212363 12,82,4862,1,1,1,1,1,1,1,2,7,22,185,16796,1
%N A212363 Number A(n,k) of Dyck n-paths all of whose ascents and descents have lengths equal to 1+k*m (m>=0); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A212363 Alois P. Heinz, <a href="/A212363/b212363.txt">Antidiagonals n = 0..140, flattened</a>
%F A212363 G.f. of column k>0 satisfies: A_k(x) = 1+A_k(x)*(x-x^k*(1-A_k(x))), g.f. of column k=0: A_0(x) = 1/(1-x).
%F A212363 A(n,k) = A(n-1,k) + Sum_{j=1..n-k} A(j,k)*A(n-k-j,k) for n,k>0; A(n,0) = A(0,k) = 1.
%e A212363 A(3,0) = 1: UDUDUD.
%e A212363 A(3,1) = 5: UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD.
%e A212363 A(4,2) = 4: UDUDUDUD, UDUUUDDD, UUUDDDUD, UUUDUDDD.
%e A212363 A(5,2) = 8: UDUDUDUDUD, UDUDUUUDDD, UDUUUDDDUD, UDUUUDUDDD, UUUDDDUDUD, UUUDUDDDUD, UUUDUDUDDD, UUUUUDDDDD.
%e A212363 A(5,3) = 4: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDUDDDD.
%e A212363 Square array A(n,k) begins:
%e A212363 1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212363 1,   1,  1,  1,  1,  1,  1,  1, ...
%e A212363 1,   2,  1,  1,  1,  1,  1,  1, ...
%e A212363 1,   5,  2,  1,  1,  1,  1,  1, ...
%e A212363 1,  14,  4,  2,  1,  1,  1,  1, ...
%e A212363 1,  42,  8,  4,  2,  1,  1,  1, ...
%e A212363 1, 132, 17,  7,  4,  2,  1,  1, ...
%e A212363 1, 429, 37, 12,  7,  4,  2,  1, ...
%p A212363 A:= proc(n, k) option remember;
%p A212363       `if`(k=0, 1, `if`(n=0, 1, A(n-1, k)
%p A212363                    +add(A(j, k)*A(n-k-j, k), j=1..n-k)))
%p A212363     end:
%p A212363 seq (seq (A(n, d-n), n=0..d), d=0..15);
%p A212363 # second Maple program
%p A212363 A:= (n, k)-> `if`(k=0, 1, coeff (series (RootOf
%p A212363               (A||k=1+A||k*(x-x^k*(1-A||k)), A||k), x, n+1), x, n)):
%p A212363 seq (seq (A(n, d-n), n=0..d), d=0..15);
%Y A212363 Columns k=0-10 give: A000012, A000108, A004148, A023432, A023427, A212364, A212365, A212366, A212367, A212368, A212369.
%K A212363 nonn,tabl,new
%O A212363 0,9
%A A212363 _Alois P. Heinz_, May 10 2012

%I A212260
%S A212260 0,1,0,1,3,5,22,64,198,710,2332,8105,28665,100653,361104,1301180,
%T A212260 4713267,17217021,63140534,232702261,861507251,3200666821,11933894310,
%U A212260 44636509320,167427781950,629691033738,2373987233286,8970240131032,33965443165016,128857452216256
%N A212260 G.f. A(x) satisfies A(x)+A(x)^2+A(x)^3 = (1-sqrt(1-4*x))/2.
%F A212260 a(n) = (sum(k=2..n, C(2*n-k-1,n-1)*(sum(i=1..k-1, (-1)^i*C(i,k-i-1) * C(k+i-1,k-1)))) +C(2*n-2,n-1))/n if n>0, a(0) = 0.
%p A212260 a:= n-> coeff (series (RootOf (A+A^2+A^3=(1-sqrt(1-4*x))/2, A), x, n+1), x, n): seq (a(n), n=0..40); _Alois P. Heinz_, May 12 2012
%o A212260 (Maxima) a(n):=(sum(binomial(2*n-k-1,n-1)*(sum((-1)^i*binomial(i,k-i-1) *binomial(k+i-1,k-1), i,1,k-1)), k,2,n) +binomial(2*n-2,n-1))/n;
%Y A212260 Cf. A103779.
%K A212260 nonn,new
%O A212260 0,5
%A A212260 _Vladimir Kruchinin_, May 12 2012

%I A210735
%S A210735 1,0,1,1,1,4,2,10,10,22,46,64,167,245,560,1035,1978,4210,7715,16497,
%T A210735 31929,65216,133295,266244,553750,1116404,2308931,4738660,9742795,
%U A210735 20204902,41622910,86539105,179358694,373018581,777157221,1618773690,3382590684,7065505631
%N A210735 Number of Dyck n-paths all of whose ascents and descents have prime lengths.
%H A210735 Alois P. Heinz, <a href="/A210735/b210735.txt">Table of n, a(n) for n = 0..900</a>
%e A210735 a(0) = 1: the empty path.
%e A210735 a(1) = 0.
%e A210735 a(2) = 1: UUDD.
%e A210735 a(3) = 1: UUUDDD.
%e A210735 a(4) = 1: UUDDUUDD.
%e A210735 a(5) = 4: UUDDUUUDDD, UUUDDDUUDD, UUUDDUUDDD, UUUUUDDDDD.
%e A210735 a(6) = 2: UUDDUUDDUUDD, UUUDDDUUUDDD.
%e A210735 a(7) = 10: UUDDUUDDUUUDDD, UUDDUUUDDDUUDD, UUDDUUUDDUUDDD, UUDDUUUUUDDDDD, UUUDDDUUDDUUDD, UUUDDUUDDDUUDD, UUUDDUUDDUUDDD, UUUUUDDDDDUUDD, UUUUUDDUUDDDDD, UUUUUUUDDDDDDD.
%e A210735 a(8) = 10: UUDDUUDDUUDDUUDD, UUDDUUUDDDUUUDDD, UUUDDDUUDDUUUDDD, UUUDDDUUUDDDUUDD, UUUDDDUUUDDUUDDD, UUUDDDUUUUUDDDDD, UUUDDUUDDDUUUDDD, UUUDDUUUDDDUUDDD, UUUUUDDDDDUUUDDD, UUUUUDDDUUUDDDDD.
%p A210735 with(numtheory):
%p A210735 b:= proc(x, y, u) option remember;
%p A210735       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, `if`(u,
%p A210735        add(b(x-ithprime(t), y, false), t=1..pi(x)),
%p A210735        add(b(x, y-ithprime(t), true ), t=1..pi(y)))))
%p A210735     end:
%p A210735 a:= n-> b(n$2, true):
%p A210735 seq (a(n), n=0..40);
%Y A210735 Cf. A210737.
%K A210735 nonn,new
%O A210735 0,6
%A A210735 _Alois P. Heinz_, May 10 2012

%I A210737
%S A210737 1,0,1,1,2,6,8,29,50,141,327,771,2047,4746,12644,30941,79886,204885,
%T A210737 522242,1365056,3505825,9185742,23907116,62636476,164624803,432540010,
%U A210737 1142827935,3017208675,7996379870,21211540268,56369770281,150086840133,400009010758
%N A210737 Number of Dyck n-paths all of whose ascents have prime lengths.
%H A210737 Alois P. Heinz, <a href="/A210737/b210737.txt">Table of n, a(n) for n = 0..700</a>
%e A210737 a(0) = 1: the empty path.
%e A210737 a(1) = 0.
%e A210737 a(2) = 1: UUDD.
%e A210737 a(3) = 1: UUUDDD.
%e A210737 a(4) = 2: UUDDUUDD, UUDUUDDD.
%e A210737 a(5) = 6: UUDDUUUDDD, UUDUUUDDDD, UUUDDDUUDD, UUUDDUUDDD, UUUDUUDDDD, UUUUUDDDDD.
%e A210737 a(6) = 8: UUDDUUDDUUDD, UUDDUUDUUDDD, UUDUUDDDUUDD, UUDUUDDUUDDD, UUDUUDUUDDDD, UUUDDDUUUDDD, UUUDDUUUDDDD, UUUDUUUDDDDD.
%p A210737 with(numtheory):
%p A210737 b:= proc(x, y, u) option remember;
%p A210737       `if`(x<0 or  y<x, 0, `if`(x=0 and y=0, 1, b(x, y-1, true)+
%p A210737       `if`(u, add (b(x-ithprime(t), y, false), t=1..pi(x)), 0)))
%p A210737     end:
%p A210737 a:= n-> b(n, n, true):
%p A210737 seq (a(n), n=0..40);
%Y A210737 Cf. A210735.
%K A210737 nonn,new
%O A210737 0,5
%A A210737 _Alois P. Heinz_, May 10 2012

%I A212268
%S A212268 0,0,1,0,2,4,14,48,432,2316,24743,223506,2537594,26771235,344781871,
%T A212268 3927465497,53273396228,660418431957
%N A212268 Number of circular permutations of 1..n with no adjacent three summing to a prime, ignoring rotations and reversals
%e A212268 Some solutions for n=8
%e A212268 ..5....6....4....2....2....5....5....6....3....3....4....2....2....5....2....2
%e A212268 ..1....1....1....1....1....1....1....1....1....1....1....1....1....1....1....1
%e A212268 ..6....7....7....7....3....8....8....7....8....6....7....7....3....6....6....5
%e A212268 ..8....2....6....6....4....3....7....8....7....8....6....8....4....7....8....4
%e A212268 ..4....5....2....8....5....7....6....5....5....2....8....6....8....3....4....6
%e A212268 ..3....3....8....4....7....2....2....3....6....5....2....4....6....4....3....8
%e A212268 ..7....4....5....3....8....6....4....4....4....7....5....5....7....2....7....7
%e A212268 ..2....8....3....5....6....4....3....2....2....4....3....3....5....8....5....3
%K A212268 nonn,new
%O A212268 1,5
%A A212268 _R. H. Hardin_ May 12 2012

%I A211371
%S A211371 0,1,1,3,7,29,131,585,3083,17089,97987,607977,3926731,26344001,
%T A211371 185908739,1358432937,10279616891,80819893393,655374770131,
%U A211371 5482528852761,47329769940331,420061520283617,3832533793409027,35926633641149865,345280194806563931
%N A211371 The number of indecomposable n-permutations that have only cycles of length 3 or less.
%F A211371 G.f.: 1-1/A(x) where A(x) is the o.g.f. for A057693.
%e A211371 a(4) = 7 because we have: 2431, 3241, 3412, 4132, 4213, 4231, 4321.
%t A211371 nn = 20; a = x + x^2/2 + x^3/3; b =
%t A211371 Total[Range[0, nn]! CoefficientList[Series[Exp[a], {x, 0, nn}], x]* x^Range[0, nn]]; CoefficientList[Series[1 - 1/b, {x, 0, nn}], x]
%Y A211371 Cf. A057693, A003319, A140456.
%K A211371 nonn,new
%O A211371 0,4
%A A211371 _Geoffrey Critzer_, May 11 2012

%I A212205
%S A212205 1,1,2,4,8,18,36,86,172,426,852,2162,4324,11166,22332,58438,116876,
%T A212205 309042,618084,1648154,3296308,8851206,17702412,47813790,95627580,
%U A212205 259585002,519170004,1415431266,2830862532,7747200558,15494401116,42545600310,85091200620,234346445154,468692890308,1294260644906,2588521289812,7165245015510,14330490031020
%N A212205 G.f.: ((1+2*x)*sqrt(1-6*x^2+x^4)-1+5*x^2-2*x^3)/(2*x*(1-6*x^2)).
%D A212205 D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.
%K A212205 nonn,new
%O A212205 0,3
%A A212205 _N. J. A. Sloane_, May 11 2012

%I A212199
%S A212199 0,1,0,1,1,5,11,39,113,377,1207,4043,13509,45957,157171,542671,
%T A212199 1884665,6586993,23137647,81662355,289414157,1029598333,3675337963,
%U A212199 13160833623,47261437761,170164260713,614154154791,2221545593179,8052506141653,29244341625077,106397352342243,387745600670175,1415284544031241,5173441096267489,18937206005320415,69409364862108451
%N A212199 G.f.: (1+3*x+4*x^2-sqrt(1-2*x-7*x^2))/(4+8*x).
%D A212199 S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted rooted non-separable planar maps, Arxiv preprint arXiv:1202.1790, 2012
%K A212199 nonn,new
%O A212199 0,6
%A A212199 _N. J. A. Sloane_, May 11 2012

%I A212259
%S A212259 0,0,0,1,10,65,385,2345,15204
%N A212259 Number of binary increasing trees with n nodes and "min-path" of length 5.
%D A212259 F. Disanto, Andre' permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, Arxiv preprint arXiv:1202.1139, 2012
%Y A212259 A diagonal of A186366.
%K A212259 nonn,more,new
%O A212259 2,5
%A A212259 _N. J. A. Sloane_, May 11 2012

%I A212258
%S A212258 0,0,1,6,25,105,490,2548,14698
%N A212258 Number of binary increasing trees with n nodes and "min-path" of length 4.
%D A212258 F. Disanto, Andre' permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, Arxiv preprint arXiv:1202.1139, 2012
%Y A212258 A diagonal of A186366.
%K A212258 nonn,more,new
%O A212258 2,4
%A A212258 _N. J. A. Sloane_, May 11 2012

%I A211602
%S A211602 0,1,3,7,20,70,287,1356,7248
%N A211602 Number of binary increasing trees with n nodes and "min-path" of length 3.
%D A211602 F. Disanto, Andre' permutations of the second kind associated to strictly binary increasing trees and left to right minima in their sub-permutations, Arxiv preprint arXiv:1202.1139, 2012
%Y A211602 A diagonal of A186366.
%K A211602 nonn,more,new
%O A211602 2,3
%A A211602 _N. J. A. Sloane_, May 11 2012

%I A210729
%S A210729 1,2,8,16,31,55,95,160,266,438,717,1169,1901,3086,5004,8108,13131,
%T A210729 21259,34411,55692,90126,145842,235993,381861,617881,999770,1617680,
%U A210729 2617480,4235191,6852703,11087927,17940664,29028626,46969326,75997989,122967353
%N A210729 a(n) = a(n-1)+a(n-2)+n+3 with n>1, a(0)=1, a(1)=2.
%H A210729 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A210729 G.f.: (1-x+4*x^2-3*x^3)/((1-x-x^2)*(1-x)^2).
%F A210729 a(n) = 3*fibonacci(n+1)+2*fibonacci(n+3)-n-6. - _Vaclav Kotesovec_, May 13 2012
%t A210729 Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,35}] (* _Vaclav Kotesovec_, May 13 2012 *)
%o A210729 (Python)
%o A210729 prpr, prev = 1,2
%o A210729 for n in range(2,99):
%o A210729     current = prev+prpr+n+3
%o A210729     print prpr,
%o A210729     prpr = prev
%o A210729     prev = current
%Y A210729 Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
%Y A210729 Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
%Y A210729 Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
%Y A210729 Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
%Y A210729 Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).
%Y A210729 Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
%Y A210729 Cf. A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.
%K A210729 nonn,easy,new
%O A210729 0,2
%A A210729 _Alex Ratushnyak_, May 10 2012

%I A212381
%S A212381 1,2,10,78,728,7848,94374,1242910,17696008,269931648,4382342380,
%T A212381 75347334440,1366637858280,26068366769840,521600687481182,
%U A212381 10924123724430678,239025078013012744,5454868593441272080,129644175575638999380,3204394919259792492432
%N A212381 G.f. satisfies: A(x) = x^2 + d/dx A(A(x))/2.
%F A212381 G.f. A(x) satisfies: A(A(x)) = Sum_{n>=4} 2*a(n-1)/n * x^n.
%F A212381 G.f. satisfies: A(x) = x^2 + A'(x)*A'(A(x))/2.
%e A212381 G.f.: A(x) = x^2 + 2*x^3 + 10*x^4 + 78*x^5 + 728*x^6 + 7848*x^7 +...
%e A212381 such that
%e A212381 A(A(x)) = x^4 + 4*x^5 + 26*x^6 + 208*x^7 + 1962*x^8 + 20972*x^9 + 248582*x^10 + 3217456*x^11 + 44988608*x^12 +...+ 2*a(n-1)*x^(n+1)/(n+1) +...
%e A212381 Related expansions:
%e A212381 A'(x) = 2*x + 6*x^2 + 40*x^3 + 390*x^4 + 4368*x^5 + 54936*x^6 +...
%e A212381 A'(A(x)) = 2*x^2 + 4*x^3 + 26*x^4 + 180*x^5 + 1640*x^6 + 17112*x^7 +...
%o A212381 (PARI) {a(n)=local(A=x^2+x^3);for(i=1,n,A=x^2+deriv(subst(A,x,A+x*O(x^n)))/2);polcoeff(A,n)}
%o A212381 for(n=2,30,print1(a(n),", "))
%K A212381 nonn,new
%O A212381 2,2
%A A212381 _Paul D. Hanna_, May 11 2012

%I A212380
%S A212380 1,1,2,6,19,65,231,849,3193,12238,47614,187578,746731,2999264,
%T A212380 12139398,49463204,202729889,835242106,3457178366,14369424882,
%U A212380 59949943825,250967906314,1053891879086,4438207804725,18739249857981,79312365115719,336430161046134
%N A212380  G.f. satisfies: A(x) = x^2 + A(A(x))/x.
%F A212380  G.f. satisfies: A(x) = ( A(x)^3 + A(A(A(x))) ) / A(A(x)).
%e A212380  G.f.: A(x) = x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 65*x^7 + 231*x^8 +...
%e A212380 such that
%e A212380 A(A(x)) = x^4 + 2*x^5 + 6*x^6 + 19*x^7 + 65*x^8 + 231*x^9 +...+ a(n+1)*x^(n+2) +...
%e A212380 Related expansions:
%e A212380 A(A(A(x))) = x^8 + 4*x^9 + 16*x^10 + 62*x^11 + 243*x^12 + 956*x^13 +...
%e A212380 A(x)^3 = x^6 + 3*x^7 + 9*x^8 + 31*x^9 + 111*x^10 + 411*x^11 + 1556*x^12 +...
%e A212380 A(x)*A(A(x)) = x^6 + 3*x^7 + 10*x^8 + 35*x^9 + 127*x^10 + 473*x^11 + 1799*x^12 +...
%e A212380 where A(x)*A(A(x)) = A(x)^3 + A(A(A(x))).
%o A212380  (PARI) {a(n)=local(A=x^2+x^3);for(i=1,n,A=x^2+subst(A,x,A+x*O(x^n))/x);polcoeff(A,n)}
%o A212380 for(n=2,40,print1(a(n),", "))
%K A212380 nonn,new
%O A212380 2,3
%A A212380 _Paul D. Hanna_, May 11 2012

%I A211216
%S A211216 1,1,2,5,14,42,132,429,1430,4862,16795,58766,207783,740924,2660139,
%T A211216 9603089,34818270,126676726,462125928,1689438278,6186432967,
%U A211216 22682699779,83249302471,305773834030,1123771473120,4131947428007,15197952958467,55915691993228
%N A211216 Expansion of (1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
%C A211216 In the paper of Kitaev, Remmel and Tiefenbruck (see Link lines), Q_(132)^(k,0,0,0)(x,0) represents a generating function depending on k and x.
%C A211216 For successive values ​​of k we have:
%C A211216 k=1, the g.f. of A000012: 1/(1-x);
%C A211216 k=2,      "      A011782: (1-x)/(1-2*x);
%C A211216 k=3,      "      A001519: (1-2*x)/(1-3*x+x^2);
%C A211216 k=4,      "      A124302: (1-3*x+x^2)/(1-4*x+3*x^2);
%C A211216 k=5,      "      A080937: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3);
%C A211216 k=6,      "      A024175: (1-5*x+6*x^2-x^3)/(1-6*x+10*x^2-4*x^3);
%C A211216 k=7,      "      A080938: (1-6*x+10*x^2-4*x^3)/(1-7*x+15*x^2-10*x^3+x^4);
%C A211216 k=8,      "      A033191: (1-7*x+15*x^2-10*x^3+x^4)/(1-8*x+21*x^2
%C A211216                            -20*x^3+5*x^4).
%C A211216 This sequence corresponds to the case k=9.
%C A211216 We observe that the coefficients of numerators and denominators are in A115139.
%C A211216 In general, Q_(132)^(k,0,0,0)(x,0) is the generating function for Dyck paths whose maximum height is less than or equal to k; also, it is the generating function of rooted binary trees T which have no nodes 'eta' such that there are >= k left edges on the path from 'eta' to the root of T (see cited paper, page 11).
%H A211216 Bruno Berselli, <a href="/A211216/b211216.txt">Table of n, a(n) for n = 0..1000</a>
%H A211216 Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012 (page 10, Corollary 3).
%H A211216 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (9,-28,35,-15,1).
%F A211216 G.f.: (1-3*x+x^2)*(1-5*x+5*x^2)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5).
%t A211216 CoefficientList[Series[(1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4)/(1 - 9 x + 28 x^2 - 35 x^3 + 15 x^4 - x^5), {x, 0, 27}], x]
%o A211216 (PARI) Vec((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)+O(x^28))
%o A211216 (MAGMA) m:=28; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-8*x+21*x^2-20*x^3+5*x^4)/(1-9*x+28*x^2-35*x^3+15*x^4-x^5)));
%Y A211216 Cf. A001519, A011782, A024175, A033191, A080937, A080938, A124302.
%K A211216 nonn,easy,new
%O A211216 0,3
%A A211216 _Bruno Berselli_, May 11 2012

%I A211180
%S A211180 1,1,1,2,3,6,10,21,40,85,170,362,752,1618,3438,7447,16065,35080,76574,
%T A211180 168424,370880,820968,1820598,4052594,9038457,20216002,45301380,
%U A211180 101746560,228918438,516016266,1165005168,2634463663,5965815375,13528669545,30717837778
%N A211180 G.f. satisfies: A(x) = x + A( x^2 + x^2*A(x) ).
%H A211180 Paul D. Hanna, <a href="/A211180/b211180.txt">Table of n, a(n) for n = 1..512</a>
%F A211180 Given g.f. A(x), define g(x) = x^2*(1 + A(x)); since A(x) = x + A(g(x))
%F A211180 then A(x) equals the sum of all iterations of g(x):
%F A211180 A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...
%e A211180 G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 21*x^8 +...
%e A211180 Define g(x) = x^2*(1 + A(x)) which satisfies: A(g(x)) = A(x) - x,
%e A211180 then the initial iterations of g(x) begin:
%e A211180 g(x) = x^2 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 6*x^8 + 10*x^9 + 21*x^10 +...
%e A211180 g(g(x)) = x^4 + 2*x^5 + 4*x^6 + 7*x^7 + 14*x^8 + 26*x^9 + 52*x^10 +...
%e A211180 g(g(g(x))) = x^8 + 4*x^9 + 12*x^10 + 30*x^11 + 73*x^12 + 170*x^13 +...
%e A211180 g(g(g(g(x)))) = x^16 + 8*x^17 + 40*x^18 + 156*x^19 + 530*x^20 +...
%e A211180 where A(x) = x + g(x) + g(g(x)) + g(g(g(x))) + g(g(g(g(x)))) +...
%o A211180 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A,x,x^2*(1+A)+x*O(x^n)));polcoeff(A,n)}
%o A211180 for(n=1,45,print1(a(n),", "))
%K A211180 nonn,new
%O A211180 1,4
%A A211180 _Paul D. Hanna_, May 11 2012

%I A210839
%S A210839 1,1,1,3,7,17,45,121,333,937,2675,7735,22613,66711,198361,593873,
%T A210839 1788697,5416097,16477471,50342595,154397465,475169249,1466983101,
%U A210839 4542061223,14100318843,43879550851,136858661589,427747723605,1339505345521,4202281084715,13205593222009
%N A210839 G.f. satisfies: A(x) = x + A( x*A(x)/(1 - x*A(x)) ).
%C A210839 Compare the g.f. to the trivial identity:
%C A210839 G(x) = x + G( x*G(x)/(1 + x*G(x)) ) which holds when G(x) = x/(1-x).
%F A210839 G.f. satisfies: A(-A(-x)) = x.
%e A210839 G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 7*x^5 + 17*x^6 + 45*x^7 + 121*x^8 +...
%e A210839 Related expansion:
%e A210839 x*A(x)/(1-x*A(x)) = x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 28*x^7 + 73*x^8 + 195*x^9 + 536*x^10 + 1501*x^11 + 4269*x^12 + 12306*x^13 + 35869*x^14 +...
%o A210839 (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A,x,x*A/(1-x*A+x*O(x^n))));polcoeff(A,n)}
%o A210839 for(n=1,30,print1(a(n),", "))
%K A210839 nonn,new
%O A210839 1,4
%A A210839 _Paul D. Hanna_, May 10 2012

%I A210736
%S A210736 1,1,1,2,3,6,10,20,35,70,126,252,462,924,1716,3432,6435,12870,24310,
%T A210736 48620,92378,184756,352716,705432,1352078,2704156,5200300,10400600,
%U A210736 20058300,40116600,77558760,155117520,300540195,601080390,1166803110,2333606220,4537567650
%N A210736 Expansion of (1 + sqrt( (1 + 2*x) / (1 - 2*x))) / 2 in powers of x.
%C A210736 Hankel transform is period 4 sequence [ 1, 0, -1, 0, ...] A056594 and the Hankel transform of sequence omitting a(0) is the all 1s sequence A000012. This is the unique sequence with that property.
%F A210736 G.f.: 2 * x / (-1 + 2*x + sqrt(1 - 4*x^2)).
%F A210736 G.f. A(x) satisfies A(x) = A(x)^2 - x / (1 - 2*x).
%F A210736 G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 / (1 - x).
%F A210736 G.f. A(x) satisfies A(1/3) = (1 + sqrt(5))/2.
%F A210736 G.f. A(x) = 1 + x / (1 - 2*x + x / A(x)).
%F A210736 G.f. A(x) = 1 + x / (1 - x / (1 - x / (1 + x / A(x)))).
%F A210736 G.f. A(x) = 1 + x * A001405(x). A000000 a(n+1) = A001405(n).
%F A210736 Convolution inverse is A210628. Partial sums is A072100.
%F A210736 Binomial transform with offset 1 is A211278 with offset 1. a(n+2) * a(n) - a(n+1)^2 = A138350(n-1).
%e A210736 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 20*x^7 + 35*x^8 + 70*x^9 + ...
%o A210736 (PARI) {a(n) = if( n<1, n==0, binomial( n - 1, (n - 1)\2))}
%o A210736 (PARI) {a(n) = polcoeff( (1 + sqrt( (1 + 2*x) / (1 - 2*x) + x * O(x^n))) / 2, n)}
%Y A210736 Cf. A001405, A056594, A138350, A210628, A211278.
%K A210736 nonn,new
%O A210736 0,4
%A A210736 _Michael Somos_, May 10 2012

%I A212333
%S A212333 1,1,25,1728,234256,52521875,17596287801,8235430000000,
%T A212333 5132188731375616,4108400332687853397,4108469075197275390625,
%U A212333 5019255990031848807858176,7355827511386641000000000000,12736801848653359358345383963927,25724477018923486959881583081626689
%N A212333 n-th power of the n-th pentagonal number.
%H A212333 Bruno Berselli, <a href="/A212333/b212333.txt">Table of n, a(n) for n = 0..100</a>
%F A212333 a(n) = A000326(n)^n.
%t A212333 Join[{1}, Table[(n ((3 n - 1)/2))^n, {n, 14}]]
%o A212333 (MAGMA) [(n*(3*n-1)/2)^n: n in [0..14]];
%Y A212333 Cf. A000312, A085741, A062206.
%K A212333 nonn,easy,new
%O A212333 0,3
%A A212333 _Bruno Berselli_, May 09 2012

%I A211919
%S A211919 0,0,1,14,46,118,267,497,871,1438,2215,3273,4713,6526,8844,11758,
%T A211919 15303,19590,24796,30862,38053,46457,56115,67206,79971,94357,110628,
%U A211919 128979,149481,172291,197776,225789,256763,290878,328179,369011
%N A211919 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>=3*y*z.
%C A211919 A211919(n)+A211920(n)=n^4.
%C A211919 See A211795 for a guide to related sequences.
%t A211919 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211919 (Do[If[w*x >= 3 y*z, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211919 Map[t[#] &, Range[0, 40]] (* A211919 *)
%t A211919 (* Peter Moses, Apr 13 2012 *)
%Y A211919 Cf. A211795.
%K A211919 nonn,new
%O A211919 0,4
%A A211919 _Clark Kimberling_, Apr 28 2012

%I A211918
%S A211918 0,0,1,6,32,100,219,441,797,1312,2061,3107,4451,6248,8526,11336,14823,
%T A211918 19090,24122,30164,37253,45501,55091,66154,78663,92979,109170,127281,
%U A211918 147629,170403,195536,223509,254329,288192,325385,366061
%N A211918 Number of (w,x,y,z) with all terms in {1,...,n} and w*x>3*y*z.
%C A211918 A211918(n)+A211912(n)=n^4.
%C A211918 See A211795 for a guide to related sequences.
%t A211918 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211918  (Do[If[w*x > 3 y*z, s = s + 1],
%t A211918  {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211918 Map[t[#] &, Range[0, 40]] (* A211918 *)
%t A211918 (* Peter Moses, Apr 13 2012 *)
%Y A211918 Cf. A211795.
%K A211918 nonn,new
%O A211918 0,4
%A A211918 _Clark Kimberling_, Apr 27 2012

%I A211917
%S A211917 0,1,15,67,210,507,1029,1904,3225,5123,7785,11368,16023,22035,29572,
%T A211917 38867,50233,63931,80180,99459,121947,148024,178141,212635,251805,
%U A211917 296268,346348,402462,465175,534990,612224,697732,791813,895043
%N A211917 Number of (w,x,y,z) with all terms in {1,...,n} and w*x<3*y*z.
%C A211917 A211917(n)+A211919(n)=n^4.
%C A211917 See A211795 for a guide to related sequences.
%t A211917 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211917 (Do[If[w*x < 3 y*z, s = s + 1],
%t A211917 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211917 Map[t[#] &, Range[0, 40]] (* A211917 *)
%t A211917 (* Peter Moses, Apr 13 2012 *)
%Y A211917 Cf. A211795.
%K A211917 nonn,new
%O A211917 0,3
%A A211917 _Clark Kimberling_, Apr 27 2012

%I A211812
%S A211812 0,1,15,75,224,525,1077,1960,3299,5249,7939,11534,16285,22313,29890,
%T A211812 39289,50713,64431,80854,100157,122747,148980,179165,213687,253113,
%U A211812 297646,347806,404160,467027,536878,614464,700012,794247,897729
%N A211812 Number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and w*x<=3*y*z.
%C A211812 A211812(n)+A211918(n)=n^4.
%C A211812 See A211795 for a guide to related sequences.
%t A211812 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211812  (Do[If[w*x <= 3 y*z, s = s + 1],
%t A211812  {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211812 Map[t[#] &, Range[0, 40]] (* A211812 *)
%t A211812 (* Peter Moses, Apr 13 2012 *)
%Y A211812 Cf. A211795.
%K A211812 nonn,new
%O A211812 0,3
%A A211812 _Clark Kimberling_, Apr 27 2012

%I A211809
%S A211809 0,0,5,23,79,188,402,742,1283,2059,3164,4633,6615,9112,12299,16253,
%T A211809 21114,26924,33932,42161,51885,63153,76182,91068,108172,127453,149257,
%U A211809 173726,201163,231591,265520,302889,344195,389508,439180,493458
%N A211809 Number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and w*x>=2*y*z.
%C A211809 A211809(n)+A211795(n)=n^4.
%C A211809 See A211795 for a guide to related sequences.
%t A211809 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211809  (Do[If[w*x >= 2 y*z, s = s + 1],
%t A211809  {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211809 Map[t[#] &, Range[0, 40]] (* A211809 *)
%t A211809 (* Peter Moses, Apr 13 2012 *)
%Y A211809 Cf. A211795.
%K A211809 nonn,new
%O A211809 0,3
%A A211809 _Clark Kimberling_, Apr 27 2012

%I A210746
%S A210746 1,1,1,1,1,3,3,3,3,3,3,5,5,7,7,7,9
%N A210746 A leaf weight sequence.
%C A210746 See Isgur et al. for precise definition.
%D A210746 ABRAHAM ISGUR, VITALY KUZNETSOV AND STEPHEN M. TANNY, A combinatorial approach for solving certain nested recursions with non-slow solutions, Arxiv preprint arXiv:1202.0276, 2012
%Y A210746 Cf. A210745.
%K A210746 nonn,more,new
%O A210746 1,6
%A A210746 _N. J. A. Sloane_, May 10 2012

%I A210745
%S A210745 1,1,1,1,1,3,3,3,3,3,3,3,5,5,7,7,9,9,9,9,9,9,9,9,9,11
%N A210745 The leaf weight sequence w_{2,3,4}.
%D A210745 ABRAHAM ISGUR, VITALY KUZNETSOV AND STEPHEN M. TANNY, A combinatorial approach for solving certain nested recursions with non-slow solutions, Arxiv preprint arXiv:1202.0276, 2012
%Y A210745 Cf. A210746.
%K A210745 nonn,more,new
%O A210745 1,6
%A A210745 _N. J. A. Sloane_, May 10 2012

%I A211679
%S A211679 6,28,6,28,496,84,270,1488,1638,84,270,1488,1638,24384,210,17360,
%T A211679 43400,284480,2229500,2886100,3780,66960,167400,406224,1097280,
%U A211679 6656832,13035330,3780,66960,167400,406224,1097280,6656832,13035330,29410290
%N A211679 Triangle of earliest friendly numbers having n friends.
%C A211679 Row n contains the earliest set of friendly numbers of length n. The sequence A211677 contains the last element of each row.
%Y A211679 Cf. A074902 (friendly numbers), A211677.
%K A211679 nonn,hard,tabl,new
%O A211679 2,1
%A A211679 _T. D. Noe_, May 10 2012

%I A212371
%S A212371 1,1,7,110,2875,109683,5678706,380631612,31942104109,3272150145947,
%T A212371 401101904099311,57890233456712428,9706532459502104648,
%U A212371 1869487973632573739154,409621529316840179292622,101253590975320030584465534,28030292175164530782257192631
%N A212371 Self-convolution yields A212370.
%C A212371 A212370 satisfies: 1 = Sum_{n>=0} A212370(n)*x^n*[Sum_{k=0..n+1} binomial(n+1, k)^2*(-x)^k]^2.
%e A212371 G.f.: A(x) = 1 + x + 7*x^2 + 110*x^3 + 2875*x^4 + 109683*x^5 +...
%e A212371 such that
%e A212371 A(x)^2 = 1 + 2*x + 15*x^2 + 234*x^3 + 6019*x^4 + 226656*x^5 +...+ A212370(n)*x^n +...
%Y A212371  Cf. A212370.
%K A212371 nonn,new
%O A212371 0,3
%A A212371 _Paul D. Hanna_, May 10 2012

%I A212370
%S A212370 1,2,15,234,6019,226656,11629128,774788698,64757369211,6615393335250,
%T A212370 809323719822671,116638942433360112,19535480098041792024,
%U A212370 3759317862736434388304,823134193681237065635088,203355215614514847510001434,56269314099500094422938613707
%N A212370  G.f.: 1 = Sum_{n>=0} a(n)*x^n * [ Sum_{k=0..n+1} binomial(n+1, k)^2*(-x)^k ]^2.
%C A212370  Compare to the g.f. G(x) for A006013(n) = C(3*n+1,n)/(n+1), which satisfies:
%C A212370 (1) 1 = Sum_{n>=0} A006013(n)*x^n*[Sum_{k=0..n+1} C(n+1,k)^2*(-x)^k]^2,
%C A212370 (2) G(x) = (1 + x*G(x)^(3/2))^2 so that G(x)^(1/2) is an integer series.
%e A212370  G.f.: A(x) = 1 + 2*x + 15*x^2 + 234*x^3 + 6019*x^4 + 226656*x^5 +...
%e A212370 Note that the square-root of the g.f., A(x)^(1/2), is an integer series:
%e A212370 A(x)^(1/2) = 1 + x + 7*x^2 + 110*x^3 + 2875*x^4 + 109683*x^5 +...+ A212371(n)*x^n +...
%o A212370  (PARI) {a(n)=if(n==0, 1, -polcoeff(sum(m=0, n-1, a(m)*x^m*sum(k=0, m+1, binomial(m+1, k)^2*(-x)^k)^2+x*O(x^n)), n))}
%o A212370 for(n=0,20,print1(a(n),", "))
%Y A212370  Cf. A212371.
%K A212370 nonn,new
%O A212370 0,2
%A A212370 _Paul D. Hanna_, May 10 2012

%I A206493
%S A206493 1,2,6,3,24,8,12,4,20,30,120,10,40,15,72,5,60,24,20,36,36,144,120,12,
%T A206493 252,48,56,18,180,84,720,6,336,72,126,28,60,24,112,42,240,42,90,168,
%U A206493 192,140,504,14,63,288,168,56,30,64,1152,21,56,210,360,96,168,840,96,7,384
%N A206493 Product, over all vertices v of the rooted tree with Matula-Goebel number n, of the number of vertices in the subtree with root v.
%C A206493 a(n) is called tree factorial. See, for example, the Brouder reference.
%C A206493 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 emenating from the root; to a tree T with r oot degree m>=2 there c orresponds the product of the Matula-Goebel numbers of the m branches of T.
%D A206493 Ch. Brouder, Runge-Kutta methods and renormalization, Eur. Phys. J. C 12, 2000, 521-534.
%D A206493 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206493 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206493 I. Gutman and Y-N. Yeh, Deducing properties of trees from t heir Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206493 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D A206493 J. Fulman, Mixing time for a random walk on rooted trees, The Electronic Journal of Combinatorics, 16, 2009, #R139.
%H A206493 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A206493 Denote by V(k) the number of vertices of the rooted tree with Matula-Goebel number k. If n is the m-th prime, then a(n) = a(m)*V(n); if n=rs, r,s>=2, then a(n) = a(r)a(s)V(n)/{V(r)V(s)}. The Maple program is based on these recurrence relations.
%e A206493 a(7)=12 because the rooted tree with Matula-Goebel number 7 is Y; denoting the vertices in preorder by a,b,c, and d, the number of vertices of the subtrees having these roots are 4, 3, 1, and 1, respectively. a(11)=120 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices; the subtrees have 5,4,3,2,1 vertices.
%p A206493 with(numtheory): 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: H := 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 V(n)*H(pi(n)) else H(r(n))*H(s(n))*V(n)/(V(r(n))*V(s(n))) end if end proc: seq(H(n), n = 1 .. 100);
%K A206493 nonn,new
%O A206493 1,2
%A A206493 _Emeric Deutsch_, May 10 2012

%I A210730
%S A210730 0,0,4,9,19,35,62,106,178,295,485,793,1292,2100,3408,5525,8951,14495,
%T A210730 23466,37982,61470,99475,160969,260469,421464,681960,1103452,1785441,
%U A210730 2888923,4674395,7563350,12237778,19801162,32038975,51840173,83879185,135719396
%N A210730 a(n) = a(n-1)+a(n-2)+n+2 with n>1, a(0)=a(1)=0.
%H A210730 Bruno Berselli, <a href="/A210730/b210730.txt">Table of n, a(n) for n = 0..1000</a>
%H A210730 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A210730 G.f.: x^2*(4-3*x)/((1-x)^2*(1-x-x^2)). [_Bruno Berselli_, May 10 2012]
%F A210730 a(n) = A210677(n)-1. [_Bruno Berselli_, May 10 2012]
%t A210730 RecurrenceTable[{a[0] == a[1] == 0, a[n] == a[n - 1] + a[n - 2] + n + 2}, a, {n, 36}] (* _Bruno Berselli_, May 10 2012 *)
%o A210730 (MAGMA) I:=[0, 0, 4, 9]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..37]]; // _Bruno Berselli_, May 10 2012
%Y A210730 Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).
%Y A210730 Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).
%Y A210730 Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.
%Y A210730 Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).
%Y A210730 Cf. A023548: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=0 (except first 2 terms).
%Y A210730 Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).
%Y A210730 Cf. A210731: a(n)=a(n-1)+a(n-2)+n+3, a(0)=a(1)=0.
%K A210730 nonn,easy,new
%O A210730 0,3
%A A210730 _Alex Ratushnyak_, May 10 2012

%I A206494
%S A206494 0,1,1,2,1,3,2,6,6,4,1,12,3,8,10,24,2,30,6,20,20,5,6,60,20,15,90,40,4,
%T A206494 60,1,120,15,10,40,180,12,30,45,120,3,120,8,30,210,36,10,360,80,140,
%U A206494 30,90,24,630,35,240,90,24,2,420,30,6,420,720,105,105,6,60,126,280,20,1260
%N A206494 Number of ways to take apart the rooted tree corresponding to the Matula-Goebel number n by sequentially removing terminal edges.
%C A206494 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 A206494 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206494 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206494 I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206494 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D A206494 J. Fulman, Mixing time for a random walk on rooted trees, The Electronic Journal of Combinatorics, 16, 2009, #R139.
%H A206494 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A206494 a(the m-th prime) = a(m); a(rs) = a(r)a(s)binom(E(rs),E(r)), where E(k) is the number of edges of the rooted tree with Matula-Goebel number k; the Maple program is based on these recurrence relations.
%F A206494 For a rooted tree T with n vertices, the desired number is n!/P, where P is the product, over all vertices v of T, of the number of vertices in the subtree with root v. See Eq. (3) in the Fulman reference.
%e A206494 a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; denoting the edges in preorder by 1,2,3, it can be taken apart either in the order 231 or 321. a(11) =1 because the rooted tree with Matula-Goebel number 11 is the path tree with 5 vertices; any path tree can be taken apart in only one way.
%p A206494 with(numtheory): E := 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 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: a := 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 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))*a(s(n))*binomial(E(n), E(r(n))) end if end proc: seq(a(n), n = 1 .. 100);
%Y A206494 Cf. A206493
%K A206494 nonn,new
%O A206494 1,4
%A A206494 _Emeric Deutsch_, May 10 2012

%I A211076
%S A211076 3,7,11,17,23,31,41,31,31,53,47,53,47,59,67,89,79,83,97,101,101,101,
%T A211076 139,113,167,127,107,151,137,149,197,157,151,149,167,239,223,229,191,
%U A211076 211,211,239,211,277,239,251,241,313,257,251,313,307,307,347,347,347
%N A211076 Least prime q such that k^p - p is not divisible by q for any positive integer k, where p is the n-th prime.
%C A211076 For p > 2 there is some prime dividing sigma(p^(p-1)) for which k^p - p is not divisible by the prime for any k. Thus a(n) exists and is at most sigma(p^(p-1)).
%H A211076 Charles R Greathouse IV, <a href="/A211076/b211076.txt">Table of n, a(n) for n = 1..1000</a>
%o A211076 (PARI) a(n)=my(p=prime(n)); forprime(q=2,default(primelimit),for(k=2,p, if(Mod(k,q)^p==p,next(2))); return(q)) \\ _Charles R Greathouse IV_, May 09 2012
%K A211076 nonn,new
%O A211076 1,1
%A A211076 _Charles R Greathouse IV_, May 09 2012

%I A211677
%S A211677 1,28,496,1638,24384,2886100,13035330,29410290
%N A211677 First number k whose value of sigma(k)/k appears n times.
%C A211677 The values of sigma(k)/k are 1, 2, 2, 8/3, 8/3, 96/35, 32/9, 32/9. Note that these values are nondecreasing. Is that always the case? In the table below, all numbers in the same row are friendly to each other.
%C A211677 5*10^8 < a(9) <= 4426793280. a(10) <= 27477725184. a(11) <= 88071903612. a(12) <= A027687(12). - _Donovan Johnson_, May 22 2012
%D A211677 Claude W. Anderson and Dean Hickerson, Problem 6020: Friendly Integers, Amer. Math. Monthly 84 (1977) 65-66.
%H A211677 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/mpn.html">Multiply Perfect Numbers</a> (sigma(k)/k is an integer)
%H A211677 Tom De Medts, <a href="http://mathoverflow.net/questions/56376">MathOverflow: Recovering n from sigma(n)/n</a>
%H A211677 Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper13.pdf">Multiply perfect numbers, Mersenne primes and effective computability</a>, Math. Ann. 226 (1977), 195-206.
%H A211677 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FriendlyNumber.html">Friendly number</a>
%H A211677 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/FriendlyPair.html">Friendly pair</a>
%H A211677 Wikipedia, <a href="http://en.wikipedia.org/wiki/Friendly_number">Friendly number</a>
%e A211677 These are the values of k such that sigma(k)/k appears n times:
%e A211677 n   k values
%e A211677 1:  1
%e A211677 2:  6, 28
%e A211677 3:  6, 28, 496
%e A211677 4:  84, 270, 1488, 1638
%e A211677 5:  84, 270, 1488, 1638, 24384
%e A211677 6:  210, 17360, 43400, 284480, 2229500, 2886100
%e A211677 7:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330
%e A211677 8:  3780, 66960, 167400, 406224, 1097280, 6656832, 13035330, 29410290
%e A211677 These numbers appear in A211679.
%Y A211677 Cf. A000203 (sigma), A211679.
%K A211677 nonn,hard,more,new
%O A211677 1,2
%A A211677 _T. D. Noe_, May 09 2012
%E A211677 a(7)-a(8) from _Donovan Johnson_, May 10 2012

%I A210726
%S A210726 1,2,4,9,22,59,169,506,1577,5078,16729,56098,190981,658442,2294164,
%T A210726 8066363,28588554,102036631,366458118,1323463507,4803734390,
%U A210726 17515357533,64128879398,235682969485,869175349090,3215674910037,11932102898252,44396448313385,165608308422048,619217961493403,2320410595131693,8713343724905902,32782954997996347,123567462151713252,466559336920866937,1764469819249171154
%N A210726 a(2)=1, a(3)=2; thereafter a(n) = 2a(n-1)-a(n-2)+A046919(n).
%C A210726 According to Riordan (1968), this is the number of possible score sequences in a tournament with n nodes, but the latter is given by A000571, which is a different sequence.
%D A210726 J. Riordan, The number of score sequences in tournaments, J. Combin. Theory, 5 (1968), 87-89.
%Y A210726 Cf. A046919.
%K A210726 nonn,new
%O A210726 2,2
%A A210726 _N. J. A. Sloane_, May 09 2012

%I A210725
%S A210725 1,1,3,1,10,16,1,41,101,125,1,196,756,1176,1296,1,1057,6607,12847,
%T A210725 16087,16807,1,6322,65794,160504,229384,257104,262144
%N A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).
%D A210725 J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
%Y A210725 Diagonals include A000248, A000949, A000950, A000951, A000272.
%K A210725 nonn,tabl,more,new
%O A210725 1,3
%A A210725 _N. J. A. Sloane_, May 09 2012

%I A210684
%S A210684 0,7,101,864,7365,63331,554839,4931118,44339730,402709395,3687732409,
%T A210684 34007530868
%N A210684 Number of primes p < 10^n such that both 2*p+1 and 4*p+1 are composite.
%F A210684 a(n) = A006880(n) - A092816(n) - A182265(n) + 1.
%F A210684 a(n) ~ 10^n / (n log 10). - _Charles R Greathouse IV_, May 11 2012
%t A210684 f[n_] := Length[Select[Range[10^n], PrimeQ[#] && !PrimeQ[2#+1] && !PrimeQ[4#+1]&]]; Table[f[n], {n,7}]
%Y A210684 Cf. A006880, A092816, A182265, A156660, A005384, A156874.
%K A210684 nonn,new
%O A210684 1,2
%A A210684 _Enrique Pérez Herrero_, May 09 2012

%I A210666
%S A210666 100,101,110,112,113,114,115,116,117,118,119,121,122,131,133,141,144,
%T A210666 151,155,161,166,171,177,181,188,191,199,200,202,211,212,220,221,223,
%U A210666 224,225,226,227,228,229,232,233,242,244,252,255,262,266,272,277,282,288
%N A210666 Near-repdigit numbers, or numbers with all like or repeated digits but one.
%C A210666 These numbers have a majority of one digit.
%C A210666 a(n) = A031955(n+81) for n <= 244.
%H A210666 Arkadiusz Wesolowski, <a href="/A210666/b210666.txt">Table of n, a(n) for n = 1..10000</a>
%t A210666 lst = {}; Do[If[SortBy[Tally[IntegerDigits[n]], Last][[-1, -1]] == IntegerLength[n] - 1, AppendTo[lst, n]], {n, 100, 288}]; lst
%t A210666 lst = {}; Do[r = Table[a, {n}]; Do[c = FromDigits@Permutations[Join[{d}, r]]; If[d == 0, c = Rest@c]; AppendTo[lst, c], {d, 0, 9}], {a, 0, 9}, {n, 2, 2}]; Drop[DeleteDuplicates@Sort@Flatten@lst, 19]
%Y A210666 Subset of A031955. Superset of A164937.
%K A210666 base,easy,nice,nonn,new
%O A210666 1,1
%A A210666 _Arkadiusz Wesolowski_, May 08 2012

%I A212272
%S A212272 0,2,9,29,67,130,224,356,533,763,1055,1420,1872,2430,3121,3985,5083,
%T A212272 6510,8416,11040,14765,20207,28359,40824,60192,90650,138969,216101,
%U A212272 339763,538618,859040,1376060,2211077,3560515,5742191,9270340,14977008,24208470,39143041
%N A212272 Fibonacci(n) + n^3.
%H A212272 Bruno Berselli, <a href="/A212272/b212272.txt">Table of n, a(n) for n = 0..1000</a>
%H A212272 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (5,-9,6,1,-3,1).
%F A212272 G.f.: x*(2-x+2*x^2-9*x^3)/((1-x-x^2)*(1-x)^4).
%t A212272 Table[Fibonacci[n] + n^3, {n, 0, 38}]
%o A212272 (PARI) for(n=0, 38, print1(fibonacci(n)+n^3", "));
%o A212272 (MAGMA) [Fibonacci(n)+n^3: n in [0..38]];
%Y A212272 Cf. A000045, A000578; A001611, A002062, A160536.
%K A212272 nonn,easy,new
%O A212272 0,2
%A A212272 _Bruno Berselli_, May 09 2012

%I A210683
%S A210683 253444777,271386581,286000489,415893013,475992773,523294549,
%T A210683 620164949,794689481,838188877,840725323,846389227,884106599,
%U A210683 884951807,908725507,941796223,952288331,971614151,1002290693,1003166771,1006976797,1053792359,1097338313,1163141201
%N A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.
%C A210683 Subsequence of A089234 which itself is a subsequence of A126771:
%C A210683 a(1) = 253444777 = A089234(417) = A126771(81526),
%C A210683 a(36) = 1998782563 = A089234(5579) = A126771(788920).
%H A210683 Zak Seidov, <a href="/A210683/b210683.txt">Table of n, a(n) for n = 1..1000</a>
%Y A210683 Cf. A089234, A126771.
%K A210683 nonn,new
%O A210683 1,1
%A A210683 _Zak Seidov_, May 09 2012

%I A210682
%S A210682 1,1,2,1,1,2,2,1,2,1,1,2,2,3,3,1,2,1,2,1
%V A210682 1,1,2,-1,1,2,2,-1,-2,1,1,2,2,3,-3,-1,-2,1,2,-1
%N A210682 Triangle read by rows: T(n,k) = coefficient of x^k in polynomial U_n(x) defined by U_1 = x, U_n = n*x^n + (1-x^n)*U_(n-1), n >= 1, 1 <= k <= n(n+1)/2.
%C A210682 T(n,m) = d(m) for m <= n (cf. A000005).
%D A210682 Uchimura, Keisuke. An identity for the divisor generating function arising from sorting theory. J. Combin. Theory Ser. A 31 (1981), no. 2, 131--135. MR0629588 (82k:05015)
%e A210682 Triangle begins:
%e A210682 1
%e A210682 1 2 -1
%e A210682 1 2 2 -1 -2 1
%e A210682 1 2 2 3 -3 -1 -2 1 2 -1
%e A210682 ...
%Y A210682 Cf. A000005.
%K A210682 sign,more,new
%O A210682 1,3
%A A210682 _N. J. A. Sloane_, May 09 2012

%I A182410
%S A182410 1,2,2,4,4,7,7,11,11,15,17,24,25,31,34,45,48,59,64,77,83,99,109,131,
%T A182410 138,164,175,204,222,252,274,317,332,385,403,466,500,563,592,674,720,
%U A182410 799,854,957,994,1131,1196,1328,1395,1551
%N A182410 Number of length sets of integer partitions of n.
%C A182410 For an integer partition n = c(1)*1 + c(2)*2 + ... + c(n)*n, construct the set of all positive c(i) occurring at least one time.
%C A182410 a(n) is the number of distincts such sets in all integer partitions of n.
%e A182410 For n=8 the 11 possible sets are {1}, {2}, {4}, {8}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3} and {2, 4}
%t A182410 Table[Length@ Union@ Map[Union@(Length /@ Split[#]) &, IntegerPartitions[n]], {n, 1, 20}]
%Y A182410 Cf. A000041 (number of partitions).
%Y A182410 Cf. A088314 (number of different ordered lists of the c(i)).
%Y A182410 Cf. A088887 (number of different sorted lists of the c(i)).
%K A182410 nonn,new
%O A182410 1,2
%A A182410 _Olivier Gérard_, May 09 2012

%I A212349
%S A212349 5,24,95,356,1259,4354
%N A212349 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(-,0,3,0)(x).
%D A212349 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212349 nonn,more,new
%O A212349 4,1
%A A212349 _N. J. A. Sloane_, May 09 2012

%I A212348
%S A212348 2,9,34,115,376,1219,3980
%N A212348 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(-,0,2,0)(x).
%D A212348 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212348 nonn,more,new
%O A212348 3,1
%A A212348 _N. J. A. Sloane_, May 09 2012

%I A212323
%S A212323 1,2,8,25,78,238,721,2174,6540,19649,58994,177058,531297,1594090,
%T A212323 4782592,14348297,43045734,129138566,387417905,1162257286,3486777636,
%U A212323 10460342257,31381041898,94143150170,282429490113,847288534418,2541865706936,7625597288569
%N A212323 3^n - Fibonacci(n).
%H A212323 Bruno Berselli, <a href="/A212323/b212323.txt">Table of n, a(n) for n = 0..1000</a>
%H A212323 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-2,-3).
%F A212323 G.f.: (1-2*x+2*x^2)/((1-3*x)*(1-x-x^2)).
%t A212323 Table[3^n - Fibonacci[n], {n, 0, 27}]
%o A212323 (PARI) for(n=0, 27, print1(3^n-fibonacci(n)", "));
%o A212323 (MAGMA) [3^n-Fibonacci(n): n in [0..27]];
%Y A212323 Cf. A000045, A000244, A099036, A212262.
%K A212323 nonn,easy,new
%O A212323 0,2
%A A212323 _Bruno Berselli_, May 09 2012

%I A212262
%S A212262 1,4,10,29,84,248,737,2200,6582,19717,59104,177236,531585,1594556,
%T A212262 4783346,14349517,43047708,129141760,387423073,1162265648,3486791166,
%U A212262 10460364149,31381077320,94143207484,282429582849,847288684468,2541865949722,7625597681405
%N A212262 3^n + Fibonacci(n).
%H A212262 Bruno Berselli, <a href="/A212262/b212262.txt">Table of n, a(n) for n = 0..1000</a>
%H A212262 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (4,-2,-3).
%F A212262 G.f.: (1-2*x)*(1+2*x)/((1-3*x)*(1-x-x^2)).
%t A212262 Table[3^n + Fibonacci[n], {n, 0, 27}]
%o A212262 (PARI) for(n=0, 27, print1(3^n+fibonacci(n)", "));
%o A212262 (MAGMA) [3^n+Fibonacci(n): n in [0..27]];
%Y A212262 Cf. A000045, A000244, A001611, A117591, A212323.
%K A212262 nonn,easy,new
%O A212262 0,2
%A A212262 _Bruno Berselli_, May 08 2012

%I A210626
%S A210626 3449,3169,2897,2633,2377,2129,1889,1657,1433,1217,1009,809,617,433,
%T A210626 257,89,71,223,367,503,631,751,863,967,1063,1151,1231,
%U A210626 1303,1367,1423,1471,1511,1543,1567,1583,1591,1591,1583,1567,1543,1511,1471
%V A210626 3449,3169,2897,2633,2377,2129,1889,1657,1433,1217,1009,809,617,433,
%W A210626 257,89,-71,-223,-367,-503,-631,-751,-863,-967,-1063,-1151,-1231,
%X A210626 -1303,-1367,-1423,-1471,-1511,-1543,-1567,-1583,-1591,-1591,-1583,-1567,-1543,-1511,-1471
%N A210626 Prime-generating polynomial: 4*n^2 - 284*n + 3449.
%C A210626 The polynomial generates 35 primes/negative values of primes in row starting from n=0.
%C A210626 The polynomial 4*n^2 + 12*n - 1583 generates the same primes in reverse order.
%C A210626 Note: we found in the same family of prime-generating polynomials (with the discriminant equal to 199*2^p, where p is odd) the polynomial 32n^2 - 944n + 6763 (with it's "reversed polynomial" 32n^2 - 976n + 7243), generating 31 primes in row and the polynomial 4n^2 - 428n + 5081 (4n^2 + 188n - 4159), generating 31 primes in row.
%H A210626 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial">MathWorld: Prime-Generating Polynomial</a>
%K A210626 sign,new
%O A210626 0,1
%A A210626 _Marius Coman_, May 08 2012

%I A182409
%S A182409 1583,1567,1543,1511,1471,1423,1367,1303,1231,1151,1063,
%T A182409 967,863,751,631,503,367,223,71,89,257,433,617,809,1009,1217,
%U A182409 1433,1657,1889,2129,2377,2633,2897,3169,3449,3737,4033,4337,4649,4969,5297,5633,5977
%V A182409 -1583,-1567,-1543,-1511,-1471,-1423,-1367,-1303,-1231,-1151,-1063,
%W A182409 -967,-863,-751,-631,-503,-367,-223,-71,89,257,433,617,809,1009,1217,
%X A182409 1433,1657,1889,2129,2377,2633,2897,3169,3449,3737,4033,4337,4649,4969,5297,5633,5977
%N A182409 Prime-generating polynomial: 4*n^2 + 12*n - 1583.
%C A182409 The polynomial generates 35 primes/negative values of primes in row starting from n=0.
%C A182409 The polynomial 4*n^2 - 284*n + 3449 generates the same primes in reverse order.
%C A182409 Other related polynomials:
%C A182409 For n = 6n+6 than n = n-11 we get 144n^2 - 2808n + 12097 which generates 16 primes in row starting from n=0 (with the discriminant equal to 2^9*3^2*199);
%C A182409 For n = 12n+12 than n = n-15 we get 576n^2 - 15984n + 109297 which generates 17 primes in row starting from n=0 (with the discriminant equal to 2^11*3^2*199).
%C A182409 So this polynomial opens at least two directions of study:
%C A182409 (1) polynomials of type 4n^2 + 12n - p, where p is prime (could be of the form 30k+23);
%C A182409 (2) polynomials with the discriminant equal to 2^n*3^m*199, where n is odd and m is even (an example of such a polynomial, with the discriminant equal to 2^5*3^4*199, is 36n^2 - 1020n + 3643 which generates 32 primes for values of n from 0 to 34).
%H A182409 E. W. Weisstein, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial">MathWorld: Prime-Generating Polynomial</a>
%K A182409 sign,new
%O A182409 0,1
%A A182409 _Marius Coman_, May 09 2012

%I A182408
%S A182408 2,7,34,743,1546,598078,6027057,10163241031,242407820869
%N A182408 Number of ways to place k non-attacking knights on an n x n toroidal chessboard, summed over all k >= 0.
%H A182408 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Non-attacking chess pieces</a>
%Y A182408 Cf. A141243, A182407, A067958, A027683, A067960.
%K A182408 nonn,hard,new
%O A182408 1,1
%A A182408 _Vaclav Kotesovec_, May 09 2012

%I A182407
%S A182407 2,9,34,982,11284,1048768,48027971,23807996588,3430123782371,
%T A182407 8141109957322587,4098570575535958632,46676507893324203092812,
%U A182407 77374614378004006943995980
%N A182407 Number of ways to place k non-attacking knights on an n x n cylindrical chessboard, summed over all k >= 0.
%H A182407 V. Kotesovec, <a href="http://web.telecom.cz/vaclav.kotesovec/math.htm">Non-attacking chess pieces</a>
%Y A182407 Cf. A141243, A182408, A172964.
%K A182407 nonn,hard,new
%O A182407 1,1
%A A182407 _Vaclav Kotesovec_, May 09 2012

%I A212347
%S A212347 14,56,144,300,550
%N A212347 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,4,0,0)(x).
%D A212347 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212347 nonn,more,new
%O A212347 5,1
%A A212347 _N. J. A. Sloane_, May 09 2012

%I A212346
%S A212346 1,1,2,5,14,28,48,75,110,154
%N A212346 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,4,0,0)(x).
%D A212346 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%F A212346 Conjecture: this appears to equal (n+3)(n^2-4)/6 for n >= 3.
%K A212346 nonn,more,new
%O A212346 0,3
%A A212346 _N. J. A. Sloane_, May 09 2012

%I A212345
%S A212345 9,18,45,126,378,1088
%N A212345 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%D A212345 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212345 nonn,more,new
%O A212345 4,1
%A A212345 _N. J. A. Sloane_, May 09 2012

%I A212344
%S A212344 5,5,10,25,70,210,660
%N A212344 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%D A212344 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212344 nonn,more,new
%O A212344 4,1
%A A212344 _N. J. A. Sloane_, May 09 2012

%I A212343
%S A212343 5,18,42,80,135,210
%N A212343 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%D A212343 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212343 nonn,more,new
%O A212343 4,1
%A A212343 _N. J. A. Sloane_, May 09 2012

%I A212161
%S A212161 6,10,23,27,40,44,57,61,74,78,91,95,108,112,125,129,142,146,159,163,
%T A212161 176,180,193,197,210,214,227,231,244,248,261,265,278,282,295,299,312,
%U A212161 316,329,333,346,350
%N A212161 Numbers 6 or 10 modulo 17.
%C A212161 A001844(N) = N^2 + (N+1)^2 = 4*A000217(N) + 1 is divisible by 17 if and only if N=a(n), n>=0. For the proof it suffices to show that only N=6 and N=10 from {0,1,..,16} satisfy A001844(N)== 0 (mod 17). Note that only primes of the form p= 4*k+1 (A002144) can be divisors of A001844 (see a W. Lang comment there giving the reference). Note also that if N^2 + (N+1)^2 == 0 (mod p), with any prime p (necessarily from A002144), then also  p-1-N satisfies this congruence. This explains why 10 = 17-1-6 is the (incongruent) companion of 6.
%C A212161 Partial sums of the sequence 6,4,13,4,13,4,13,4,13,4,13,... (see the o.g.f., and subtract 6 to see the remaining 4, 13=17-4 periodicity).
%F A212161 Bisection: a(2*n) = 17*n + 6, a(2*n+1) = 17*n + 10, n>=0.
%F A212161 O.g.f.: (6+4*x+7*x^2)/((1-x)*(1-x^2)).
%F A212161 a(n) = 1/4*(34*n+9*(-1)^n+15). - _Paolo P. Lava_, May 16 2012
%e A212161 Divisibility of A001844 by 17:
%e A212161 n=0: A001844(6) = 85 = 5*17 == 0 (mod 17).
%e A212161 n=2: A001844(23) = 1105 = 5*13*17 == 0 (mod 17).
%e A212161 However, 8^2 + 9^2 = 145 == 9 (mod 17) is not divisible by 17 because 8 is not a member of the present sequence.
%Y A212161 Cf. A047219 (p=5), A212160 (p=13).
%K A212161 nonn,easy,new
%O A212161 0,1
%A A212161 _Wolfdieter Lang_, May 09 2012

%I A212342
%S A212342 1,2,5,9,14,20,27,35,44
%N A212342 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,3,0,0)(x).
%D A212342 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212342 nonn,more,new
%O A212342 0,2
%A A212342 _N. J. A. Sloane_, May 09 2012

%I A211798
%S A211798 2,12,1,36,7,1,80,23,7,1,150,54,22,7,1,252,103,51,22,7,1,392,175,97,
%T A211798 50,22,7,1,576,276,164,95,50,22,7,1,810,409,258,162,95,50,22,7,1,1100,
%U A211798 579,382,254,161,95,50,22,7,1,1452,791,541,375,253,161,95,50,22
%N A211798 Rectangular array, by antidiagonals:  R(k,n) = number of ordered triples (w,x,y) such that x and y are in {1,...,n} and w^k<=x^k+y^k.
%C A211798 Limiting row:  A002412
%C A211798 row 1:  2*A002411
%C A211798 row 2:  A211791
%C A211798 row 3:  A211792
%F A211798 R(k,n)=sum{sum{floor((x^k+y^k)^(1/k)), 1<=x<=n, 1<=y<=n}}.
%e A211798 Northwest corner:
%e A211798 2...12...36...80...150...252...392
%e A211798 1...7....23...54...103...175...276
%e A211798 1...7....22...51...97....164...258
%e A211798 1...7....22...50...95....162...254
%e A211798 1...7....22...50...95....161...254
%e A211798 1...7....22...50...95....161...253
%t A211798 f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
%t A211798 t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
%t A211798 Table[t[1, n], {n, 1, 45}]  (* 2*A002411 *)
%t A211798 Table[t[2, n], {n, 1, 45}]  (* A211791 *)
%t A211798 Table[t[3, n], {n, 1, 45}]  (* A211792 *)
%t A211798 TableForm[Table[t[k, n], {k, 1, 12},
%t A211798                  {n, 1, 16}]] (* A211798 *)
%t A211798 Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]
%K A211798 nonn,tabl,new
%O A211798 1,1
%A A211798 _Clark Kimberling_, Apr 26 2012

%I A211797
%S A211797 0,0,1,15,55,156,338,666,1159,1905,2950,4399,6275,8748,11847,15721,
%T A211797 20450,26228,33078,41271,50799,61947,74832,89674,106480,125659,147291,
%U A211797 171618,198767,229139,262698,300007,340983,386100,435544,489598
%N A211797 Number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and w*x>2*y*z.
%C A211797 A211797(n)+A211875(n)=n^4.
%C A211797 See A211795 for a guide to related sequences.
%t A211797 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211797     (Do[If[w*x > 2 y*z, s = s + 1], {w, 1, #},
%t A211797       {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A211797 Map[t[#] &, Range[0, 40]] (* A211797 *)
%t A211797 (* Peter Moses, Apr 13 2012 *)
%Y A211797 Cf. A211795.
%K A211797 nonn,new
%O A211797 0,4
%A A211797 _Clark Kimberling_, Apr 27 2012

%I A211787
%S A211787 0,1,15,66,201,469,958,1735,2937,4656,7050,10242,14461,19813,26569,
%T A211787 34904,45086,57293,71898,89050,109201,132534,159424,190167,225296,
%U A211787 264966,309685,359823,415889,478142,547302,623514,707593,799821
%N A211787 Number of (w,x,y,z) with all terms in {1,...,n} and w*x<=2*y*z.
%C A211787 A211787(n)+A21197(n)=n^4.
%C A211787 See A211795 for a guide to related sequences.
%t A211787 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A211787     (Do[If[w*x <= 2 y*z, s = s + 1],
%t A211787     {w, 1, #}, {x, 1, #}, {y, 1, #},
%t A211787       {z, 1, #}] &[n]; s)]];
%t A211787 Map[t[#] &, Range[0, 40]] (* A211787 *)
%t A211787 (* Peter Moses, Apr 13 2012 *)
%Y A211787 Cf. A211795.
%K A211787 nonn,new
%O A211787 0,3
%A A211787 _Clark Kimberling_, Apr 27 2012

%I A182404
%S A182404 11,12,14,16,21,23,25,32,38,41,49,52,56,58,61,65,83,85,94,101,102,104,
%T A182404 106,110,111,113,119,120,131,133,137,140,146,160,164,166,173,179,191,
%U A182404 197,199,201,203,205,210,223,229,230,232,250,289,292,298,302,308
%N A182404 Numbers whose digit sum as well as sum of the squares of the digits is a prime.
%C A182404 Note that the cube analogue "Numbers whose digit sum as well as sum of the cubes of the digits is a prime" only occurs when A007953(n) = Digital sum (i.e. sum of digits) of n) = 2, as otherwise A055012(n) = Sum of cubes of digits of n = 2, i.e. n = 2, 11, 20, 101, 110, 1001, 1010, ... since for natural numbers A^3 + B^3 is divisible by A+B. Hence "Numbers whose digit sum as well as sum of the cubes of the digits is a prime" begins 2, 11, 101, ... [Jonathan Vos Post, May 10 2012].
%e A182404 25 is here because 2 + 5 = 7 and 2*2 + 5*5 = 29 both are prime.
%t A182404 fQ[n_] := Module[{d = IntegerDigits[n]}, PrimeQ[Total[d]] && PrimeQ[Total[d^2]]]; Select[Range[500], fQ] (* _T. D. Noe_, May 09 2012 *)
%Y A182404 Cf. A108662.
%K A182404 nonn,base,new
%O A182404 1,1
%A A182404 Sumit Maheshwari, May 09 2010

%I A212341
%S A212341 14,56,188,603,1907
%N A212341 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,0,4,0)(x).
%D A212341 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212341 nonn,more,new
%O A212341 5,1
%A A212341 _N. J. A. Sloane_, May 09 2012

%I A212340
%S A212340 1,1,2,5,14,28,62,143,331,738
%N A212340 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,0,4,0)(x).
%D A212340 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212340 nonn,more,new
%O A212340 0,3
%A A212340 _N. J. A. Sloane_, May 09 2012

%I A212160
%S A212160 2,10,15,23,28,36,41,49,54,62,67,75,80,88,93,101,106,114,119,127,132,
%T A212160 140,145,153,158,166,171,179,184,192,197,205,210,218,223,231,236,244,
%U A212160 249,257,262,270,275,283,288,296,301,309,314,322,327,335,340,348
%N A212160 Numbers congruent 2 or 10 modulo 13.
%C A212160 A001844(N) = N^2 + (N+1)^2 = 4*A000217(N) + 1 is divisible by 13 if and only if N=a(n), n>=0. For the proof it suffices to show that only N=2 and N=10 from {0,1,..,12} satisfy A001844(N)== 0 (mod 13). Note that only primes of the form p= 4*k+1 (A002144) can be divisors of A001844 (see a W. Lang comment there giving the reference).
%C A212160 Partial sums of the sequence [2,5,8,5,8,5,8,5,8,...] (see the o.g.f., and subtract 2 to see the 5,8 periodicity).
%F A212160 Bisection: a(2*n) = 13*n + 2, a(2*n+1) = 13*n + 10, n>=0.
%F A212160 O.g.f.: (2+8*x+3*x^2)/((1-x)*(1-x^2)).
%F A212160 a(n) = 1/4*(26*n-3*(-1)^n+11). - _Paolo P. Lava_, May 16 2012
%e A212160 Divisibility of A001844 by 13:
%e A212160 n=0: A001844(2) = 13 == 0 (mod 13).
%e A212160 n=3: A001844(23) = 1105 = 85*13 == 0 (mod 13).
%e A212160 However, 8^2 + 9^2 = 145 == 2 (mod 13) is not divisible by 13 because 8 is not a member of the present sequence.
%Y A212160 Cf. A047219 (case p=5).
%K A212160 nonn,easy,new
%O A212160 0,1
%A A212160 _Wolfdieter Lang_, May 09 2012

%I A212339
%S A212339 5,19,61,188,532,1387
%N A212339 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,0,3,0)(x).
%D A212339 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212339 nonn,more,new
%O A212339 4,1
%A A212339 _N. J. A. Sloane_, May 09 2012

%I A212338
%S A212338 2,7,21,53,124,273,577
%N A212338 Sequence of coefficients in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).
%D A212338 S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, Arxiv preprint arXiv:1201.6243, 2012.
%K A212338 nonn,more,new
%O A212338 3,1
%A A212338 _N. J. A. Sloane_, May 09 2012

%I A210678
%S A210678 1,1,6,12,24,43,75,127,212,350,574,937,1525,2477,4018,6512,10548,
%T A210678 17079,27647,44747,72416,117186,189626,306837,496489,803353,1299870,
%U A210678 2103252,3403152,5506435,8909619,14416087,23325740,37741862,61067638,98809537,159877213,258686789,418564042
%N A210678 a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.
%Y A210678 Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
%Y A210678 Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
%Y A210678 Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
%Y A210678 Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
%Y A210678 Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
%Y A210678 Cf. A210677: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.
%K A210678 nonn,easy,new
%O A210678 0,3
%A A210678 _Alex Ratushnyak_, May 09 2012

%I A210677
%S A210677 1,1,5,10,20,36,63,107,179,296,486,794,1293,2101,3409,5526,8952,14496,
%T A210677 23467,37983,61471,99476,160970,260470,421465,681961,1103453,1785442,
%U A210677 2888924,4674396,7563351,12237779,19801163,32038976,51840174,83879186,135719397,219598621,355318057
%N A210677 a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=1.
%Y A210677 Cf. A081659: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=1 (except first 2 terms and sign).
%Y A210677 Cf. A001924: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=1 (except first 4 terms).
%Y A210677 Cf. A000126: a(n)=a(n-1)+a(n-2)+n-2, a(0)=a(1)=1 (except first term).
%Y A210677 Cf. A066982: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=1.
%Y A210677 Cf. A030119: a(n)=a(n-1)+a(n-2)+n,   a(0)=a(1)=1.
%Y A210677 Cf. A210678: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=1.
%K A210677 nonn,easy,new
%O A210677 0,3
%A A210677 _Alex Ratushnyak_, May 09 2012

%I A210675
%S A210675 0,1,7,15,30,54,94,159,265,437,716,1168,1900,3085,5003,8107,13130,
%T A210675 21258,34410,55691,90125,145841,235992,381860,617880,999769,1617679,
%U A210675 2617479,4235190,6852702,11087926,17940663,29028625,46969325,75997988,122967352,198965380
%N A210675 a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.
%Y A210675 Cf. A210673: a(n)=a(n-1)+a(n-2)+n-4, a(0)=0,a(1)=1.
%Y A210675 Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0,a(1)=1 (except the first term).
%Y A210675 Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0,a(1)=1.
%Y A210675 Cf. A001924: a(n)=a(n-1)+a(n-2)+n,   a(0)=0,a(1)=1.
%Y A210675 Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0,a(1)=1.
%Y A210675 Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0,a(1)=1.
%Y A210675 Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0,a(1)=1.
%K A210675 nonn,easy,new
%O A210675 0,3
%A A210675 _Alex Ratushnyak_, May 09 2012

%I A212337
%S A212337 1,8,42,184,731,2736,9844,34448,118101,398584,1328606,4384392,
%T A212337 14348911,46633952,150663528,484275616,1549681961,4939611240,
%U A212337 15690529810,49686677720,156905298051,494251688848,1553362450652,4871909504304,15251194969981,47659984281176
%N A212337 Expansion of 1/(1-4*x+3*x^2)^2.
%H A212337 Bruno Berselli, <a href="/A212337/b212337.txt">Table of n, a(n) for n = 0..1000</a>
%H A212337 S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I,</a> arXiv:1201.6243v1 [math.CO], 2012. See (16).
%H A212337 <a href="/index/Rec#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).
%F A212337 G.f.: 1/((1-x)^2*(1-3*x)^2). [_Bruno Berselli_, May 11 2012]
%F A212337 a(n) = 1+n*(1+9*3^n)/4. [_Bruno Berselli_, May 11 2012]
%t A212337 Table[1 + n ((1 + 9 3^n)/4), {n, 0, 25}] (* _Bruno Berselli_, May 11 2012 *)
%o A212337 (MAGMA) m:=26; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)^2*(1-3*x)^2)));
%K A212337 nonn,easy,new
%O A212337 0,2
%A A212337 _N. J. A. Sloane_, May 09 2012

%I A210673
%S A210673 0,1,1,1,2,2,2,1,1,5,12,24,44,77,131,219,362,594,970,1579,2565,
%T A210673 4161,6744,10924,17688,28633,46343,74999,121366,196390,317782,514199,
%U A210673 832009,1346237,2178276,3524544,5702852,9227429,14930315,24157779,39088130,63245946
%V A210673 0,1,-1,-1,-2,-2,-2,-1,1,5,12,24,44,77,131,219,362,594,970,1579,2565,
%W A210673 4161,6744,10924,17688,28633,46343,74999,121366,196390,317782,514199,
%X A210673 832009,1346237,2178276,3524544,5702852,9227429,14930315,24157779,39088130,63245946
%N A210673 a(n)=a(n-1)+a(n-2)+n-4, a(0)=0, a(1)=1.
%Y A210673 Cf. A066982: a(n)=a(n-1)+a(n-2)+n-2, a(0)=0, a(1)=1 (except the first term).
%Y A210673 Cf. A104161: a(n)=a(n-1)+a(n-2)+n-1, a(0)=0, a(1)=1.
%Y A210673 Cf. A001924: a(n)=a(n-1)+a(n-2)+n,   a(0)=0, a(1)=1.
%Y A210673 Cf. A192760: a(n)=a(n-1)+a(n-2)+n+1, a(0)=0, a(1)=1.
%Y A210673 Cf. A192761: a(n)=a(n-1)+a(n-2)+n+2, a(0)=0, a(1)=1.
%Y A210673 Cf. A192762: a(n)=a(n-1)+a(n-2)+n+3, a(0)=0, a(1)=1.
%Y A210673 Cf. A210675: a(n)=a(n-1)+a(n-2)+n+4, a(0)=0, a(1)=1.
%K A210673 sign,easy,new
%O A210673 0,5
%A A210673 _Alex Ratushnyak_, May 09 2012

%I A210680
%S A210680 0,1,3,8,27,89,272,873,2859,8936,25491,67665,228192,752241,2167859,
%T A210680 6833896,18464907,52758345,142005584,440498361,1186119547,3461957320,
%U A210680 9357060899,26968655777,72945663424,226371206881,613739200867,1752444795592,4702791627067,14623717009785
%N A210680 a(n) = (3*a(n-1)) XOR a(n-2).
%F A210680 a(n) = (3*a(n-1)) XOR a(n-2), a(0)=0, a(1)=1.
%Y A210680 Cf. A168081: a(n) = (2*a(n-1)) XOR a(n-2), a(0)=0, a(1)=1.
%Y A210680 Cf. A101624: a(n) = (2*a(n-2)) XOR a(n-1), a(0)=0, a(1)=1.
%Y A210680 Cf. A204771: a(n) = (3*a(n-2)) XOR a(n-1), a(0)=0, a(1)=1.
%K A210680 nonn,base,easy,new
%O A210680 0,3
%A A210680 _Alex Ratushnyak_, May 09 2012

%I A212284
%S A212284 20,21,12,40,50,21,70,153,10,190,108,40,126,135,50,153,162,20,180,190,
%T A212284 70,207,216,80,234,243,30,261,270,190,290,594,102,315,324,40,342,351,
%U A212284 120,370,380,130,792,405,50,423,432,150,450,460,160,480,490,60,504
%N A212284 First a(n) > 1 such that its sum of digits is the same in base 10 as in base n.
%C A212284 There might exist an n for which there is no solution, in which case a(n) would be set to 0 by convention; however, no such case was found so far. Problem: does it exist?
%H A212284 Stanislav Sykora, <a href="/A212284/b212284.txt">Table of n, a(n) for n = 2..10000</a>
%e A212284 a(12)=108 because 108 is the first number >1 such that when written in base 10 and in base 12 (i.e., 90), the sum of the expansion digits is the same, namely 9.
%Y A212284 Cf. A037308.
%K A212284 nonn,easy,new
%O A212284 2,1
%A A212284 _Stanislav Sykora_, May 08 2012

%I A212283
%S A212283 2,6,4,6,12,21,8,10,20,12,14,172,30,46,16,18,36,20,22,126,46,24,26,
%T A212283 126,28,30,58,60,120,126,32,34,68,36,38,185,78,40,42,126,44,46,90,92,
%U A212283 138,48,50,246,52,54,106,108,56,58,114,60,62,120,182,126,188,378
%N A212283 First a(n) > 1 such that its sum of digits is the same in base 2 as in base n
%C A212283 Theoretically, there might exist an n for which there is no solution, in which case a(n) would be set to 0 by convention; however, no such case was found so far. Problem: does it exist?
%H A212283 Stanislav Sykora, <a href="/A212283/b212283.txt">Table of n, a(n) for n = 2..10000</a>
%e A212283 Example: a(13) = 172 because 172 is the first number >1 such that its expansions in base 2 (10101100) and in base 13 (103) have the same sum of digits, namely 4.
%Y A212283 Cf. A037301, A212222.
%K A212283 nonn,new
%O A212283 2,1
%A A212283 _Stanislav Sykora_, May 08 2012

%I A211771
%S A211771 1,4,6,8,9,12,14,15,16,18,24,25,26,27,28,34,35,36,38,39,45,46,48,49,
%T A211771 56,57,58,68,69,78,123,124,125,126,128,129,134,135,136,138,145,146,
%U A211771 147,148,156,158,159,168,169,178,189,234,235,236,237,238,245,246,247
%N A211771 Nonprime numbers with distinct digits in ascending order.
%C A211771 Sequence is finite with 411 terms, last term is a(411) = 123456789.
%C A211771 Complement of A052015 with respect to A009993. Supersequence of A211772.
%H A211771 Jaroslav Krizek, <a href="/A211771/b211771.txt">Table of n, a(n) for n = 1..411</a> (complete list)
%F A211771 A178788(a(n)) = 1.
%Y A211771 Cf. A052015 (primes with distinct digits in ascending order), A009993 (numbers with distinct digits in ascending order), A211772 (nonprime numbers all of whose divisors are numbers whose decimal digits are in ascending order).
%K A211771 nonn,base,new
%O A211771 1,2
%A A211771 _Jaroslav Krizek_, May 07 2012

%I A211822
%S A211822 1,361,703,1369,1387,2071,2413,2701,3097,3439,3781,4033,4699,5149,
%T A211822 5329,5833,6031,6697,6859,7201,7363,7543,7957,8227,9253,9271,9937,
%U A211822 10027,10279,10963,11359,11647,11881,11899,11989,13213,13357
%N A211822 Nonprime numbers with all divisors with additive digital root of 1.
%C A211822 Complement of A061237 (prime numbers == 1 (mod 9)) with respect to A211821.
%e A211822 Number 6859 with divisors 1, 19, 361, 6859 is in sequence because all divisors have additive digital root of 1.
%t A211822 (* First run the program for A211821 *) Select[A211821, Not[PrimeQ[#]] &] (* Alonso del Arte, May 02 2012 *)
%Y A211822 Cf. A211821, A024906, A061237, A017173, A211823.
%K A211822 nonn,base,new
%O A211822 1,2
%A A211822 _Jaroslav Krizek_, Apr 26 2012

%I A211821
%S A211821 1,19,37,73,109,127,163,181,199,271,307,361,379,397,433,487,523,541,
%T A211821 577,613,631,703,739,757,811,829,883,919,937,991,1009,1063,1117,1153,
%U A211821 1171,1279,1297,1369,1387,1423,1459,1531,1549,1567,1621,1657,1693,1747,1783
%N A211821 Numbers with all divisors with additive digital root of 1.
%C A211821 All divisors of numbers from this sequence are in this sequence. Likewise, the product of any terms in this sequence is a number that is also in this sequence.
%C A211821 Union of A061237 (prime numbers == 1 (mod 9)) and nonprime numbers A211822.
%C A211821 Subsequence of A017173 (numbers of form 9n+1). - Krizek
%C A211821 For prime numbers, it is enough to verify that the number itself is congruent to 1 mod 9. The first composite term is 361, which is the square of the first prime in this sequence. - Alonso del Arte, May 02 2012
%F A211821 a(n) = 9*k(n) + 1 for k(n) = A211823(n).
%e A211821 Number 703 with divisors 1, 19, 37, 703 is in sequence because all divisors have additive digital root of 1.
%t A211821 digitalRoot[n_, b_:10] := FixedPoint[Plus@@IntegerDigits[#, b] &,  n]; A211821 = Select[Range[1, 1999, 9], Union[digitalRoot[Divisors[#]]] == {1} &] (* Alonso del Arte, May 02 2012 *)
%Y A211821 Cf. A211822, A211823, A024906, A061237, A017173.
%K A211821 nonn,base,new
%O A211821 1,2
%A A211821 _Jaroslav Krizek_, Apr 26 2012

%I A206492
%S A206492 1,2,1,3,2,1,3,2,1,4,3,2,1,4,3,2,1,4,3,3,1,4,3,3,1,5,4,3,2,1,5,4,3,2,
%T A206492 1,5,4,3,2,1,5,4,4,3,1,5,4,4,3,1,5,4,4,3,1,6,5,4,3,2,1,5,4,6,4,1,5,4,
%U A206492 4,3,1,6,5,5,5,3,1,5,4,6,4,1,6,5,5,4,3,1,6,5,5,4,3,1,6,5,4,3,2,1,6,5,5,5,3,1,6,5,7,7,4,1,7,6,5,4,3,2,1,6,5,5,5,3,1,7,6,6,7,6,3,1,6,5,6,6,4,1,6,5,5,4,3
%N A206492 Irregular triangle read by rows: T(n,k) is the number of subtrees with k nodes in the rooted tree having Matula-Goebel number n.
%C A206492 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 emenating from the root; to a tree T with r oot degree m>=2 there c orresponds the product of the Matula-Goebel numbers of the m branches of T.
%C A206492 Number of entries in row n = A061775(n).
%C A206492 Sum of entries in row n = A184161(n).
%C A206492 For the number of subtrees containing the root, see A206491.
%D A206492 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206492 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206492 I. Gutman and Y-N. Yeh, Deducing properties of trees from t heir Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206492 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A206492 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%e A206492 Row 3 is 3,2,1 because the rooted tree with Matula-Goebel number 3 is the path tree  a - b - c, having 3 subtrees with 1 node each (a, b, c), 2 subtrees with 2 nodes each (ab, bc), and 1 subtree with 3 nodes (abc).
%p A206492 with(numtheory): 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: R := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 and k = 1 then 1 elif bigomega(n) = 1 then R(pi(n), k-1) else add(R(r(n), j)*R(s(n), k+1-j), j = 1 .. k) end if end proc: W := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 then W(pi(n), k)+R(n, k) else W(r(n), k)+W(s(n), k)+R(n, k)-R(r(n), k)-R(s(n), k) end if end proc: for n to 20 do seq(W(n, k), k = 1 .. V(n)) end do; # yields sequence in triangular form
%Y A206492 Cf. A061775, A184161, A206491.
%K A206492 nonn,tabf,new
%O A206492 1,2
%A A206492 _Emeric Deutsch_, May 08 2012

%I A206491
%S A206491 1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,2,1,1,3,3,1,1,2,3,2,1,1,2,2,2,
%T A206491 1,1,1,1,1,1,1,3,4,3,1,1,1,2,2,1,1,2,3,3,1,1,2,3,3,2,1,1,4,6,4,1,1,1,
%U A206491 1,2,1,1,3,5,5,3,1,1,1,3,3,1,1,3,4,4,3,1,1,2,4,4,3,1,1,2,2,2,2,1,1,1,2,3,2,1,1,4,7,7,4,1
%N A206491 Irregular triangle read by rows: T(n,k) is the number of root subtrees with k nodes in the rooted tree having Matula-Goebel number n.
%C A206491 A root subtree of a rooted tree G is a subtree of G containing the root.
%C A206491 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 emenating from the root; to a tree T with r oot degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%C A206491 Number of entries in row n = A061775(n).
%C A206491 Sum of entries in row n = A184160(n).
%C A206491 For the number of all subtrees of a given size, see A206492.
%D A206491 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206491 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206491 I. Gutman and Y-N. Yeh, Deducing properties of trees from t heir Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206491 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A206491 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%e A206491 Row 7 is 1,1,2,1 because the rooted tree with Matula-Goebel number 7 is Y; its five root subtrees have 1, 2, 3, 3, and 4 nodes.
%p A206491 with(numtheory): 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: R := proc (n, k) 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 and k = 1 then 1 elif n = 1 and 1 < k then 0 elif bigomega(n) = 1 and k = 1 then 1 elif bigomega(n) = 1 then R(pi(n), k-1) else add(R(r(n), j)*R(s(n), k+1-j), j = 1 .. k) end if end proc: for n to 40 do seq(R(n, k), k = 1 .. V(n)) end do; # yields sequence  in triangular form
%Y A206491 Cf. A061775, A184160, A206492
%K A206491 nonn,tabf,new
%O A206491 1,8
%A A206491 _Emeric Deutsch_, May 08 2012

%I A206490
%S A206490 0,2,6,6,14,14,9,9,24,24,24,19,19,19,38,12,19,29,12,31,31,38,29,24,54,
%T A206490 29,36,24,31,45,38,15,54,31,47,34,24,24,45,38,29,36,24,47,54,36,45,29,
%U A206490 38,61,47,36,15,41,74,29,38,45,31,52,34,54,43,18,63,63,24,38,54,54,38,39,36,34,70,29,65,52
%N A206490 The eccentric connectivity index of the rooted tree with Matula-Goebel number n.
%C A206490 The Matula-Goebel number of a rooted tree can be defined in the f ollowing 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 emenating from the root; to a tree T with r oot degree m>=2 there c orresponds the product of the Matula-Goebel numbers of the m branches of T.
%C A206490 The eccentric connectivity index of a simple connected graph G is defined as the sum over all vertices i of G of the product E(i) D(i), where E(i) is the eccentricity and D(i) is the degree of  vertex i.
%D A206490 F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A206490 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A206490 I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A206490 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D A206490 V. Sharma, R. Goswami, and A. K. Madan, Eccentric Connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci., 37, 1997, 273-282.
%H A206490 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv1111.4288.
%F A206490 Explanation of the Maple program: "V" finds recursively the number of vertices (needed later); "d" finds recursively the distance matrix; "a" finds the adjacency matrix from the distance matrix; "RS" finds the vector of the row sums of any matrix (will be applied to the adjacency matrix to yield the vertex degrees); "MX" finds the vector of the largest row entries of any matrix (will be applied to the distance matrix to yield the vertex eccentricities); "ECI" finds the eccentric connectivity index by taking the dot product of the two vectors just mentioned.
%e A206490 a(7)=9 because the rooted tree with Matula-Goebel number 7 is Y; the 3 pendant vertices have degree 1 and eccentricity 2 and the 4th vertex has degree 3 and eccentricity 1; 1*2 + 1*2 + 1*2 + 3*1 = 9.
%p A206490 with(numtheory): 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, dt, drs: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: dt := proc (i, j) if i = 1 and j = 1 then 0 elif i = 1 and 1 < j then 1+dd[pi(n)][1, j-1] elif 1 < i and j = 1 then 1+dd[pi(n)][i-1, 1] elif 1 < i and 1 < j then dd[pi(n)][i-1, j-1] else  end if end proc: drs := proc (i, j) if 1 <= i and 1 <= j and i <= V(r(n)) and j <= V(r(n)) then dd[r(n)][i, j] elif 1+V(r(n)) <= i and 1+V(r(n)) <= j and i <= V(n) and j <= V(n) then dd[s(n)][i-V(r(n))+1, j-V(r(n))+1] elif 1 <= i and i <= V(r(n)) and 1+V(r(n)) <= j and j <= n then dd[r(n)][i, 1]+dd[s(n)][1, j-V(r(n))+1] else dd[r(n)][1, j]+dd[s(n)][i-V(r(n))+1, 1] end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), dt) else Matrix(V(n), V(n), drs) end if end proc: for n to 1000 do dd[n] := d(n) end do: > a := proc (n) local ddd, aa: ddd := proc (n) options operator, arrow: d(n) end proc: aa := proc (i, j) if ddd(n)[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(ddd(n)), RowDimension(ddd(n)), aa) end proc: > RS := proc (m) local dim: dim := RowDimension(m): Matrix(1, dim, [seq(add(m[i, j], j = 1 .. dim), i = 1 .. dim)]) end proc: > MX := proc (m) local dim: dim := RowDimension(m): Matrix(1, dim, [seq(max(seq(m[i, j], j = 1 .. dim)), i = 1 .. dim)]) end proc: > ECI := proc (n) options operator, arrow: MatrixMatrixMultiply(RS(a(n)), Transpose(MX(d(n))))[1, 1] end proc: seq(ECI(n), n = 1 .. 77);
%K A206490 nonn,new
%O A206490 1,2
%A A206490 _Emeric Deutsch_, May 08 2012

%I A211681
%S A211681 2,3,5,7,23,37,53,73,237,373,537,737,2373,3737,5373,7373,23737,37373,
%T A211681 53737,73737,237373,373737,537373,737373,2373737,3737373,5373737,
%U A211681 7373737,23737373,37373737,53737373,73737373,237373737,373737373,537373737,737373737
%N A211681 Numbers such that all the substrings of length <= 2 are primes.
%C A211681 The terms are primes for n= 1, 2, 3, 4, 5, 6, 7, 8, 10, 21, 23, 27, 31, 43, 45, 60, 67, 82, 91,.... The further terms with index 92, 93, 94, 96, 99 are composite. For the subsequence with prime terms see A211682.
%H A211681 Hieronymus Fischer, <a href="/A211681/b211681.txt">Table of n, a(n) for n = 1..100</a>
%F A211681 a(1+8*k) = 2*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
%F A211681 a(2+8*k) = 3*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
%F A211681 a(3+8*k) = 5*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
%F A211681 a(4+8*k) = 7*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
%F A211681 a(5+8*k) = 23*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
%F A211681 a(6+8*k) = 37*10^(2k) + 37*sum_{j=0..k-1} 10^(2j),
%F A211681 a(7+8*k) = 53*10^(2k) + 73*sum_{j=0..k-1} 10^(2j),
%F A211681 a(8+8*k) = 73*10^(2k) + 73*sum_{j=0..k-1} 10^(2j), for k>=0.
%F A211681 a(n)=((2*n+7) mod 8 + d(n+4) - d(n+3))*10^d(n-1) + floor((37+36*(d(n+2)-d(n+1))*10^d(n-1)/99), where d(n)= floor(n/4).
%F A211681 Recursion for n>8:
%F A211681 a(n) = 10*(1+a(n-4)) - a(n-4) mod 10.
%F A211681 G.f.: g(x)=(2x(1+x^10) +3x^2(1+x^3+x^5+x^6) +5x^3(1+x^6) +7x^4(+x^2))/((1-10x^4)(1-x^8)).
%e A211681 a(11)=537, since all substrings of length <= 2 are primes (5, 3, 7, 53 and 37).
%e A211681 a(21)=237373, the substrings of length <= 2 are 2, 3, 7, 23, 37, 73.
%Y A211681 Cf. A019546, A035232, A039996, A046034, A085823, A211682.
%K A211681 base,nonn,new
%O A211681 1,1
%A A211681 _Hieronymus Fischer_, Apr 30 2012

%I A210668
%S A210668 1,1,2,5,16,60,260,1260,6744,39303
%N A210668 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> cba where a<b<c.
%H A210668 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920"> Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%e A210668 Contribution from _Alois P. Heinz_, May 16 2012: (Start)
%e A210668 a(3) = 5: {123, 321}, {132}, {213}, {231}, {312}.
%e A210668 a(4) = 16: {1234, 1432, 3214}, {1243, 4213}, {1324}, {1342, 4312}, {1423}, {2134, 2431}, {2143}, {2314}, {2341, 4123, 4321}, {2413}, {3124, 3421}, {3142}, {3241}, {3412}, {4132}, {4231}. (End)
%Y A210668 Cf. A210667, A210669, A210671, A212417.
%K A210668 nonn,new
%O A210668 0,3
%A A210668 _Tom Roby_, May 08 2012
%E A210668 Definition improved by _Tom Roby_, May 15 2012
%E A210668 a(0)-a(2), a(9) from _Alois P. Heinz_, May 16 2012

%I A210671
%S A210671 1,1,2,3,4,5,8,11,20,29,57
%N A210671 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb <--> bac <--> cba, where a<b<c.
%H A210671 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%K A210671 nonn,new
%O A210671 0,3
%A A210671 _Tom Roby_, May 08 2012
%E A210671 Definition improved by _Tom Roby_, May 15 2012

%I A210669
%S A210669 1,1,2,4,8,14,27,68,159,496
%N A210669 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb <--> cba where a<b<c.
%C A210669 Also number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> bac <--> cba where a<b<c.
%H A210669 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920">Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%e A210669 Contribution from _Alois P. Heinz_, May 19 2012: (Start)
%e A210669 a(3) = 4: {123, 132, 321}, {213}, {231}, {312}.
%e A210669 a(4) = 8: {1234, 1243, 1324, 1342, 1423, 1432, 3214, 4213, 4312}, {2134, 2143, 2341, 2431, 4123, 4132, 4321}, {2314}, {2413}, {3124, 3142, 3421}, {3241}, {3412}, {4231}. (End)
%Y A210669 Cf. A210667, A210668, A210671, A212417, A212418.
%K A210669 nonn,new
%O A210669 0,3
%A A210669 _Tom Roby_, May 08 2012
%E A210669 Definition improved and comment added by _Tom Roby_, May 15 2012

%I A210667
%S A210667 1,1,2,5,16,62,284,1507,9104,61766
%N A210667 Number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> acb where a<b<c.
%C A210667 Also number of equivalence classes of S_n under transformations of positionally adjacent elements of the form abc <--> bac where a<b<c.
%H A210667 Steven Linton, James Propp, Tom Roby, and Julian West, <a href="http://arxiv.org/abs/1111.3920"> Equivalence classes of permutations under various relations generated by constrained transpositions, 2011</a> arXiv:1111.3920 [math.CO]
%e A210667 Contribution from _Alois P. Heinz_, May 16 2012: (Start)
%e A210667 a(3) = 5: {123, 132}, {213}, {231}, {312}, {321}.
%e A210667 a(4) = 16: {1234, 1243, 1324, 1423}, {1342, 1432}, {2134, 2143}, {2314}, {2341, 2431}, {2413}, {3124, 3142}, {3214}, {3241}, {3412}, {3421}, {4123, 4132}, {4213}, {4231}, {4312}, {4321}. (End)
%Y A210667 Cf. A210668, A210669, A210671, A212417.
%K A210667 nonn,new
%O A210667 0,3
%A A210667 _Tom Roby_, May 08 2012
%E A210667 Definition improved by _Tom Roby_, May 15 2012
%E A210667 a(0)-a(2), a(9) from _Alois P. Heinz_, May 16 2012

%I A211667
%S A211667 0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%T A211667 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%U A211667 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N A211667 Number of iterations sqrt(sqrt(sqrt(...(n)...))) such that the result is < 2.
%C A211667 Different from A001069, but equal for n < 256.
%F A211667 a(2^(2^n)) = a(2^(2^(n-1)))+1, for n>=1.
%F A211667 G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(2^(2^k)). The explicit first terms of the g.f. are
%F A211667 g(x)=(x^2+x^4+x^16+x^256+x^65536+…)/(1-x).
%e A211667 a(n)=1, 2, 3, 4, 5 for n=2^1, 2^2, 2^4, 2^8, 2^16 =2, 4, 16, 256, 65536.
%Y A211667 Cf. A001069, A010096, A211662, A211668, A211670.
%K A211667 base,nonn,new
%O A211667 1,4
%A A211667 _Hieronymus Fischer_, Apr 30 2012

%I A211670
%S A211670 0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A211670 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,
%U A211670 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N A211670 Number of iterations (...f_4(f_3(f_2(n))))...) such that the result is < 2, where f_j(x):=x^(1/j).
%C A211670 Different from A001069, but equal for n < 16.
%F A211670 a(2^(n!)) = a(2^((n-1)!))+1, for n>1.
%F A211670 G.f.: g(x)= 1/(1-x)*sum_{k=1..infinity} x^(2^(k!)). The explicit first terms of the g.f. are
%F A211670 g(x)=(x^2+x^4+x^64+x^16777216+...)/(1-x).
%e A211670 a(n)=1, 2, 3, 4, 5 for n=2^(1!), 2^(2!), 2^(3!), 2^(4!), 2^(5!) =2, 4, 64, 16777216, 16777216^5.
%Y A211670 Cf. A001069, A010096, A084558, A211662, A211664, A211666, A211668, A211669.
%K A211670 base,nonn,new
%O A211670 1,4
%A A211670 _Hieronymus Fischer_, Apr 30 2012

%I A211665
%S A211665 1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A211665 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A211665 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A211665 Number of iterations log_10(log_10(log_10(...(n)...))) such that the result is < 1.
%C A211665 Different from A055642 and A138902.
%F A211665 With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n))))...))); E_{i=1..0} := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we get:
%F A211665 a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10)+1, for n>=1.
%F A211665 G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(E_{i=1..k} 10).
%F A211665 The explicit first terms of the g.f. are
%F A211665 g(x)=(x+ x^10+x^(10^10)+…)/(1-x).
%e A211665 a(n)=1, 2, 3, 4 for n=1, 10, 10^10, 10^(10^10) =1, 10, 10000000000, 10^10000000000.
%Y A211665 Cf. A001069, A010096, A211661, A211663, A211666, A211668, A211670.
%K A211665 base,nonn,new
%O A211665 1,10
%A A211665 _Hieronymus Fischer_, Apr 30 2012

%I A211662
%S A211662 0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A211662 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A211662 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A211662 Number of iterations log_3(log_3(log_3(...(n)...))) such that the result is < 2.
%F A211662 With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n))))...))); E_{i=1..0} := 1; example: E_{i=1..4} 3 = 3^(3^(3^3)) = 3^(3^27), we get:
%F A211662 a(E_{i=1..n} 3) = a(E_{i=1..n-1} 3)+1, for n>=1.
%F A211662 G.f.: g(x)= 1/(1-x)*sum_{k=1..infinity} x^(E_{i=1..k} b(i,k)), where b(i,k)=3 for i<k and b(i,k)=2 for i=k. The explicit first terms of the g.f. are
%F A211662 g(x)=(x^2+x^9+x^19683+…)/(1-x).
%e A211662 a(n)=0, 1, 2, 3, 4, for n=1, 2, 3^2, 3^3^2, 3^3^3^2 =1, 2, 9, 19683, 3^19683
%Y A211662 Cf. A001069, A010096, A211661, A211664, A211666, A211668, A211669.
%K A211662 base,nonn,new
%O A211662 1,9
%A A211662 _Hieronymus Fischer_, Apr 30 2012

%I A211668
%S A211668 0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A211668 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A211668 2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3
%N A211668 Number of iterations sqr(sqr(sqr(...(n)...))) such that the result is < 3.
%C A211668 For a general definition like
%C A211668 Number of iterations f(f(f(...(n)...))) such that the result is < q, , where f(x)=x^(1/p)” where p>1, q>1, the resulting g.f. is
%C A211668 g(x)= 1/(1-x)*sum_{k=0..infinity} x^(q^(p^k)). The explicit first terms of the g.f. are
%C A211668 g(x)=(x^q+x^(q^p) +x^(q^(p^2))+x^(q^(p^3))+…)/(1-x).
%F A211668 a(3^(2^n)) = a(3^(2^(n-1)))+1, for n>=1.
%F A211668 G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(3^(2^k)). The explicit first terms of the g.f. are
%F A211668 g(x)=(x^3+x^9+x^81+x^6561+x^43946721+…)/(1-x).
%e A211668 a(n)=1, 2, 3, 4, 5 for n=3^1, 3^2, 3^4, 3^8, 3^16 =3, 9, 81, 6561, 43946721.
%Y A211668 Cf. A001069, A010096, A211662, A211666, A211670.
%K A211668 base,nonn,new
%O A211668 1,9
%A A211668 _Hieronymus Fischer_, Apr 30 2012

%I A211663
%S A211663 1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%T A211663 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,
%U A211663 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3
%N A211663 Number of iterations log(log(log(...(n)...))) such that the result is < 1, where log is the natural logarithm.
%C A211663 For n<16 same as A211661.
%F A211663 With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n))))...))); E_{i=1..0} := 1; example: E_{i=1..4} 3 = 3^(3^(3^3)) = 3^(3^27), we get:
%F A211663 a(ceiling(E_{i=1..n} e)) = a(ceiling(E_{i=1..n-1} e))+1, for n>=1.
%F A211663 G.f.: g(x)= 1/(1-x)*sum_{k=0..infinity} x^(ceiling(E_{i=1..k} e)). The explicit first terms of the g.f. are
%F A211663 g(x)=(x+x^3+x^16+x^3814280+...)/(1-x).
%e A211663 a(n)=1, 2, 3, 4, for n=1, ceiling(e), ceiling(e^e), ceiling(e^e^e), =1, 3, 16, 3814280.
%Y A211663 Cf. A001069, A010096, A211662, A211664, A211666, A211668, A211669.
%K A211663 base,nonn,new
%O A211663 1,3
%A A211663 _Hieronymus Fischer_, Apr 30 2012

%I A211666
%S A211666 0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A211666 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A211666 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A211666 Number of iterations log_10(log_10(log_10(...(n)...))) such that the result is < 2.
%C A211666 Different from A004216, A057427 and A185114.
%C A211666 For a general definition like "Number of iterations log_p(log_p(log_p(...(n)...))) such that the result is < q", where p > 1, q > 0, the resulting g.f. is
%C A211666 g(x) = 1/(1-x)*sum_{k=1..infinity} x^(E_{i=1..k} b(i,k)), where b(i,k)=p for i<k and b(i,k)=q for i=k. The explicit first terms of the g.f. are
%C A211666 g(x) = (x^q+x^(p^q)+x^(p^p^q)+x^(p^p^p^q)+…)/(1-x).
%F A211666 With the exponentiation definition E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n))))...))); E_{i=1..0} c := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we get:
%F A211666 a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10)+1, for n>=1.
%F A211666 G.f.: g(x)= 1/(1-x)*sum_{k=1..infinity} x^(E_{i=1..k} b(i,k)), where b(i,k)=10 for i<k and b(i,k)=2 for i=k.
%F A211666 The explicit first terms of the g.f. are
%F A211666 g(x)=(x^2+x^100+x^(10^100)+…)/(1-x).
%e A211666 a(n)=0, 1, 2, 3 for n=1, 2, 10^2, 10^10^2 =1, 2, 100, 10^100.
%Y A211666 Cf. A001069, A010096, A211662, A211664, A211668, A211670.
%K A211666 base,nonn,new
%O A211666 1
%A A211666 _Hieronymus Fischer_, Apr 30 2012

%I A182194
%S A182194 2,4,3,9,8,7,6,5,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,100,
%T A182194 99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,
%U A182194 76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54
%N A182194 a(1)=2, a(n)=a(n-1)^2 if the minimal natural number > 1 not encountered so far is greater than a(n-1), else a(n)=a(n-1)-1.
%C A182194 A reordering of the natural numbers > 1.
%C A182194 The sequence is quasi self-inverse in that a(a(n-1)-1)=n.
%H A182194 Hieronymus Fischer, <a href="/A182194/b182194.txt">Table of n, a(n) for n = 1..10200</a>
%F A182194 a(n)=a(n-1)-1, if a(n-1)-1 > 1 is not in the set {a(k)| 1<=k<=n-1}, else a(n)=a(n-1)^2.
%F A182194 a(a(n)-1)=n+1.
%F A182194 If we define b(1)=2, b(2)=3, b(k)=b(k-2)^2+1, we get the sequence 2, 3, 5, 10, 26, 101, 677, 10202, 458330, 104080805, …. The b(k) are those terms a(n) of the original sequence for which a(n+1)=a(n)^2.
%F A182194 With these b(k) we obtain for k>1:
%F A182194 a(b(k)-2)=b(k-1),
%F A182194 a(b(k)-1)=b(k-1)^2.
%F A182194 a(b(k))=b(k-1)^2 - 1.
%F A182194 a(n)=b(m)+b(m-1)-n-2, where m is the least index such that b(m)>n+1 (valid for n>=1).
%e A182194 a(2)=4=a(1)^2, since 3>2=a(1) is the minimal number not encountered so far (because of a(1)=2);
%e A182194 a(15)=19=a(14)-1, since the minimal number not encountered so far (=10) is <=a(14)=20.
%e A182194 a(10^4)=b(8)+b(7)-10^4-2=877.
%e A182194 a(10^6)=b(10)+b(9)-10^6-2= 103539133.
%Y A182194 Cf. A020703, A038722, A132666, A132664, A132665, A132674, A210882.
%K A182194 nonn,new
%O A182194 1,1
%A A182194 _Hieronymus Fischer_, Apr 30 2012

%I A182586
%S A182586 0,2,6,8,7,3,5,4,14,16,19,21,20,9,25,10,29,31,11,13,12,33,37,40,15,41,
%T A182586 45,47,17,50,18,49,22,55,58,60,23,63,61,24,26,69,71,70,27,76,28,77,32,
%U A182586 30,82,84,87,89,34,92,90,35,97,36,39,99,38,103,105,108,110,109
%N A182586 a(n) is the unique non-negative integer m such that the Grundy function of the position [n,m] for the Wythoff game evaluates to 1.
%p A182586 mex:=proc(S) local s:
%p A182586 for s from 0 while member(s,S) do od:
%p A182586 s:
%p A182586 end:
%p A182586 GW:=proc(a,b) local i:
%p A182586 option remember:
%p A182586 mex({seq( GW(a-i,b),i=1..a), seq(GW(a,b-i),i=1..b),
%p A182586 seq(GW(a-i,b-i),i=1..min(a,b))}):
%p A182586 end:
%p A182586 W1:=proc(i) local b:
%p A182586 for b from 0 while GW(i,b)<>1 do od:
%p A182586 b:
%p A182586 end:
%p A182586 #W1seq(N): list L of length N such that [i,L[i]] is the
%p A182586 #unique position with grundy function value 1.
%p A182586 W1seq:=proc(N) local i:
%p A182586 [seq(W1(i),i=1..N)]:
%p A182586 end:
%K A182586 nonn,new
%O A182586 1,2
%A A182586 _John Y. Kim_, May 06 2012

%I A211772
%S A211772 1,4,6,8,9,12,14,15,16,18,24,25,26,27,28,34,35,36,38,39,45,46,48,49,
%T A211772 56,57,58,68,69,78,125,134,135,136,138,145,158,169,178,235,237,245,
%U A211772 247,259,267,268,278,289,356,358,469,478,578,1345,1357,1369,2479,2569
%N A211772 Nonprime numbers all of whose divisors are numbers whose decimal digits are in ascending order.
%C A211772 Sequence is finite with 63 terms, last term is a(63) = 134689.
%C A211772 Complement of A052015 with respect to A190218. Subsequence of A211771.
%H A211772 Jaroslav Krizek, <a href="/A211772/b211772.txt">Table of n, a(n) for n = 1..63</a> (complete list).
%e A211772 Divisors of 24589: 1, 67, 367, 24589 (all divisors with digits in ascending order).
%Y A211772 Cf. A052015 (primes with distinct digits in ascending order), A190218 (numbers all of whose divisors are numbers whose decimal digits are in ascending order), A211771 (nonprime numbers with distinct digits in ascending order).
%K A211772 nonn,base,fini,new
%O A211772 1,2
%A A211772 _Jaroslav Krizek_, May 07 2012

%I A211792
%S A211792 1,7,22,51,97,164,258,382,541,741,982,1271,1611,2008,2466,2986,3577,
%T A211792 4241,4982,5807,6715,7714,8808,10000,11297,12701,14217,15848,17600,
%U A211792 19477,21482,23620,25895,28313,30879,33592,36460,39487,42678,46036
%N A211792 Number of ordered triples (w,x,y) such that x and y are in {1,...,n} and w^3<=x^3+y^3.
%C A211792 Row 3 of A211798.
%F A211792 Sum{sum{floor((x^3+y^3)^(1/3)), 1<=x<=n, 1<=y<=n}}.
%e A211792 The 7 triples counted by a(2): (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).
%t A211792 f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
%t A211792 t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
%t A211792 Table[t[1, n], {n, 1, 45}]  (* 2*A002411 *)
%t A211792 Table[t[2, n], {n, 1, 45}]  (* A211791 *)
%t A211792 Table[t[3, n], {n, 1, 45}]  (* A211792 *)
%t A211792 TableForm[Table[t[k, n], {k, 1, 12},
%t A211792                  {n, 1, 16}]] (* A211798 *)
%t A211792 Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]
%Y A211792 Cf. A211798.
%K A211792 nonn,new
%O A211792 1,2
%A A211792 _Clark Kimberling_, Apr 26 2012

%I A211791
%S A211791 1,7,23,54,103,175,276,409,579,791,1050,1360,1724,2149,2640,3198,3832,
%T A211791 4543,5337,6217,7192,8265,9437,10716,12103,13609,15231,16978,18857,
%U A211791 20869,23018,25307,27745,30337,33084,35992,39066,42309,45728
%N A211791 Number of ordered triples (w,x,y) such that x and y are in {1,...,n} and w^2<=x^2+y^2.
%C A211791 Row 2 of A211798.
%F A211791 Sum{sum{floor((x^2+y^2)^(1/2)), 1<=x<=n, 1<=y<=n}}.
%e A211791 The 7 triples counted by a(2): (1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,2), (2,2,1), (2,2,2).
%t A211791 f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
%t A211791 t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
%t A211791 Table[t[1, n], {n, 1, 45}]  (* 2*A002411 *)
%t A211791 Table[t[2, n], {n, 1, 45}]  (* A211791 *)
%t A211791 Table[t[3, n], {n, 1, 45}]  (* A211792 *)
%t A211791 TableForm[Table[t[k, n], {k, 1, 12},
%t A211791                  {n, 1, 16}]] (* A211798 *)
%t A211791 Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]
%Y A211791 Cf. A211798.
%K A211791 nonn,new
%O A211791 1,2
%A A211791 _Clark Kimberling_, Apr 26 2012

%I A211811
%S A211811 0,3,11,26,52,93,149,222,314,431,577,750,952,1185,1461,1776,2132,2529,
%T A211811 2971,3468,4018,4621,5277,5988,6772,7619,8531,9512,10562,11699,12913,
%U A211811 14204,15574,17031,18585,20226,21956,23779,25707,27738,29874
%N A211811 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^3>x^3+y^3.
%C A211811 Row 3 of A182259; see A211790 for a discussion and guide to related sequences.
%t A211811 (See the program at A182259.)
%Y A211811 Cf. A211790, A182259.
%K A211811 nonn,new
%O A211811 1,2
%A A211811 _Clark Kimberling_, Apr 22 2012

%I A211808
%S A211808 1,5,1,16,5,1,36,16,5,1,69,36,16,5,1,117,69,38,16,5,1,184,119,73,38,
%T A211808 16,5,1,272,190,123,75,38,16,5,1,385,282,194,131,75,38,16,5,1,525,399,
%U A211808 290,204,131,75,38,16,5,1,696,547,415,300,210,131,75,38,16,5,1
%N A211808 Rectangular array:  R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^k<=x^k+y<k.
%C A211808 Row 1:  A055232
%C A211808 Row 2:  A211806
%C A211808 Row 3:  A211807
%C A211808 Limiting row sequence: A000330
%C A211808 Let R be the array in A211808 and let R' be the array in A182259.  Then R(k,n)+R'(k,n)=3^(n-1).
%C A211808 See the Comments at A211790.
%e A211808 Northwest corner:
%e A211808 1...5...16...36...69...117...184
%e A211808 1...5...16...36...69...119...190
%e A211808 1...5...16...38...73...123...194
%e A211808 1...5...16...38...75...131...204
%e A211808 1...5...16...38...75...131...210
%t A211808 z = 48;
%t A211808 t[k_, n_] := Module[{s = 0},
%t A211808    (Do[If[2 w^k <= x^k + y^k, s = s + 1],
%t A211808        {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
%t A211808 Table[t[1, n], {n, 1, z}]  (* A055232 *)
%t A211808 Table[t[2, n], {n, 1, z}]  (* A211806 *)
%t A211808 Table[t[3, n], {n, 1, z}]  (* A211807 *)
%t A211808 TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
%t A211808 Flatten[Table[t[k, n - k + 1],
%t A211808      {n, 1, 12}, {k, 1, n}]] (* A211808 *)
%t A211808 Table[k (4 k^2 - 3 k + 5)/6,
%t A211808      {k, 1, z}] (* row-limit sequence, A174723 *)
%t A211808 (* Peter Moses, Apr 13 2012 *)
%Y A211808 Cf. A211790.
%K A211808 nonn,tabl,new
%O A211808 1,2
%A A211808 _Clark Kimberling_, Apr 22 2012

%I A211810
%S A211810 0,3,11,28,56,97,153,230,330,453,605,788,1002,1251,1541,1872,2244,
%T A211810 2667,3139,3662,4240,4877,5571,6330,7158,8055,9021,10062,11182,12379,
%U A211810 13657,15026,16480,18021,19661,21398,23232,25169,27211,29362,31618
%N A211810 Number of ordered triples (w,x,y) with all terms in {1,...,n} and 2w^2>x^2+y^2.
%C A211810 Row 2 of A182259; see A211790 for a discussion and guide to related sequences.
%t A211810 (See the program at A182259.)
%Y A211810 Cf. A211790, A182259.
%K A211810 nonn,new
%O A211810 1,2
%A A211810 _Clark Kimberling_, Apr 22 2012

%I A182369
%S A182369 8,6,7,5,3,0,9,0,1,9,8,1,6,8,5,4,0,9,7,5,5,8,2,7,5,2,2,4,9,6,1,4,3,1,
%T A182369 8,3,8,4,4,0,2,9,7,2,3,1,3,2,8,1,1,6,9,3,7,7,1,5,6,5,8,9,5,6,1,7,6,0,
%U A182369 6,0,3,9,0,3,5,9,1,8,9,7,8,3,5,4,0,3,1,2,6,0,6,4,5,9,5,0,5,4,2,7,9,7,1,3,6,8,9,8
%N A182369 Decimal expansion of (7^(e - 1/e) - 9)*Pi^2, also known as Jenny's constant.
%C A182369 First few digits reproduce the digits of the phone number in the song "867-5309/Jenny" performed by Tommy Tutone.
%C A182369 The next digit is a 0, and the following 4 digits (1, 9, 8, 1) are the year the song was recorded (1981). (Noticed by Rob Johnson of the explainxkcd.com forums)
%H A182369 Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers-12.html#lb867_530">867.5309019816854...</a>
%H A182369 Randall Munroe, <a href="http://xkcd.com/1047/">xkcd: A Table of Slightly Wrong Equations and Identities Useful for Approximations and/or Trolling Teachers</a>
%H A182369 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JennysConstant.html">Jenny's Constant</a>
%H A182369 explain xkcd (website with unknown owner), <a href="http://www.explainxkcd.com/2012/04/25/approximations/#comments">Explain xkcd: Approximations</a> (discussion forum)
%H A182369 Wikipedia, <a href="http://en.wikipedia.org/wiki/867-5309/Jenny">867-5309/Jenny</a>
%F A182369 (7^(e - 1/e) - 9)*Pi^2.
%e A182369 867.530901981685409755827522496143183844029723...
%Y A182369 Cf. A104175.
%K A182369 new,cons,nonn
%O A182369 3,1
%A A182369 _Eric W. Weisstein_, Apr 26 2012
%E A182369 Title edited by Matthew Vandermast, May 05 2012

%I A208494
%S A208494 2,2,3,7,10,13,13,13,31,37,37,37,61,61,61,83,83,83,127,127,127,127,
%T A208494 127,179,179,179,179,179,193,193,193,193,193,193,193,193,277,277,277,
%U A208494 277,277,277,383,383,383,383,383,479,479,479,479,479,479,479,541,541,541,541,541,541,541,541,541,641,641,641,641,641,641,641,641,641,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,877,1013,1013,1013,1013,1013,1013,1013,1013,1013,1013,1013
%N A208494 Least integer m>1 such that those k! mod m with k=1,...,n are pairwise distinct
%C A208494 On Feb. 27, 2012 Zhi-Wei Sun conjectured that a(n) is a prime with the only exception a(5)=10, and that a(n) does not exceed n^2/2 for all n=2,3,4,... He also conjectured that max{n>0: 1!,...,n! are pairwise incongrurnt mod p} is asymptotically equivalent to sqrt(p), where p is an odd prime.
%C A208494 He guessed that if we replace k! in the definition of a(n) by (-1)^k*k! then a(n) is a prime with the only exception a(3)=6. If we replace k! in the definition of a(n) by (2k)! or (-1)^k*(2k)!, then Zhi-Wei Sun conjectured that a(n) will take only prime values.
%C A208494 He also has similar conjectures involving (r*k)! or (-1)^k*(r*k)! with r>2.
%H A208494 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1202&amp;L=nmbrthry&amp;T=0&amp;P=749">A function taking only prime values</a>, a message to Number Theory List, Feb. 21, 2012.
%H A208494 Zhi-Wei Sun, <a href="/A208494/b208494.txt">Table of n, a(n) for n = 1..400</a>
%e A208494 For n=5 we have a(5)=10 since 1!=1, 2!=2, 3!=6, 4!=24 and 5!=120 are pairwise incongruent mod 10 but not pairwise incongruent modulo any of 2,3,...,9.
%t A208494 R[n_,i_]:=Union[Table[Mod[k!,i],{k,1,n}]]
%t A208494 Do[Do[If[Length[R[n,i]]==n,Print[n," ",i];Goto[aa]],{i,2,Max[n^2,2]}];
%t A208494 Print[n];Label[aa];Continue,{n,1,1000}]
%Y A208494 A000142, A207982
%K A208494 new,nonn,nice
%O A208494 1,1
%A A208494 _Zhi-Wei Sun_, Feb 27 2012

%I A185001
%S A185001 5,8,10,16,22,26,106,110,182,234,282,288,318,434,766,1056,1072,1462,
%T A185001 1550,1930,3024,4330,5424,9398,10634,53094,90602,151632,384002,511638,
%U A185001 530102,1364850,1887006,2193072,3138096,6470672,6959070
%N A185001 Numbers n with the property that their basins (as defined in A204539) are 2.
%C A185001 This sequence follows on from A204540, which listed seven values of n for which basin(n) =1.  There are 37 known values of n for which basin(n) =2.  A search of numbers up to 10,000,000 has not uncovered any further integers satisfying this requirement. The possibility that there are even larger numbers with basins equal to 2 cannot be completely ruled out, but the chances of one being discovered are remote, in view of the fact that the average basin size for large values of n is approximately n/3, i.e. over 2000000 in the region where the last known such integer was discovered.
%C A185001 For unknown reasons, all integers >5 with basins equal to 1 or 2 are even.
%Y A185001 A204539, A204540
%K A185001 new,nonn,nice
%O A185001 1,1
%A A185001 _Colm Fagan_, Jan 23 2012

%I A204810
%S A204810 1,2,3,4,5,8,9,14,16,18,33,48,52,96,97,184,185,368,369,384,385,472,473
%N A204810 Langton's ant, symmetrical pattern
%C A204810 The sequence shows the number of steps that Langton's ant needs to create symmetrical patterns.
%H A204810 Clemens Hovekamp, <a href="http://hovekamp.info/clemens/langton-ameise/symmetrien-langton-ameise.pdf">Langton Ameise, symmetrische Muster</a>
%H A204810 Wikipedia, <a href="http://en.wikipedia.org/wiki/Langton%27s_ant">Langton's ant</a>
%K A204810 new,nonn,full,fini
%O A204810 1,2
%A A204810 _Clemens Hovekamp_, Jan 19 2012

%I A200919
%S A200919 0,0,0,1,3,5,10,13,19,25
%N A200919 Number of crossings on periodic braids with n strands such that all strands meet.
%H A200919 Duraid Madina, <a href="/A200919/a200919.txt">Illustration of initial terms, n = 1..9</a>
%K A200919 new,nonn,hard
%O A200919 1,5
%A A200919 _Duraid Madina_, Nov 24 2011

%I A204420
%S A204420 1,0,1,0,6,3,0,120,90,15,0,5040,4620,1260,105,0,362880,378000,132300,
%T A204420 18900,945,0,39916800,45571680,18711000,3534300,311850,10395,0,
%U A204420 6227020800,7628100480,3511347840,794593800,94594500,5675670,135135
%N A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into k cycles of even length, k=0..n.
%C A204420 The row polynomials t(n,x):=sum(T(n,k)*x^k,k=0..n) satisfy the recurrence relation t(n,x)= (2n-1)(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2)
%C A204420 T(n,1):=(2n-1)!
%F A204420 T(n,k) = (2n-1)!!*2^{n-k]*A132393(n,k).
%F A204420 T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
%F A204420 T(n,n):=(2n-1)!!, 0<=k<=n.
%e A204420 [1], [0,1], [0,6, 3], [0,120, 90, 15], [0,5040, 4620, 1260, 105], [0,362880, 378000, 132300, 18900, 945],
%e A204420 [0,39916800, 45571680, 18711000, 3534300, 311850, 10395],
%p A204420 T_row := proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)*coeff(expand(pochhammer(x, n)), x, k), k=0..n) end:
%Y A204420 Cf. A049218, A060523, A060524, A132393, A048994.
%K A204420 new,easy,nonn,tabl
%O A204420 0,5
%A A204420 _José H. Nieto S._, Jan 15 2012

%I A192983
%S A192983 1,4,24,264,5640,151200,5722920,282868992,18371308032,1504791561600,
%T A192983 148978034686800,18007146260231040,2528615024682544512,
%U A192983 426310052282058252672,81830910530970671616000,18305445786667543107072000,4570435510076312321728158720
%N A192983 a(n) is the number of pairs (g, h) of elements of the symmetric group S_n such that g and h have conjugates that commute.
%C A192983 a(n) / n!^2 is the probability that two permutation in S_n, chosen independently and uniformly at random, have conjugates that commute.
%C A192983 Apparently n | a(n), and, for  n>1,  n*(n-1) | a(n). - Alexander R. Povolotsky, Sep 30 2011
%D A192983 SIMON R. BLACKBURN, JOHN R. BRITNELL, AND MARK WILDON, THE PROBABILITY THAT A PAIR OF ELEMENTS OF A FINITE GROUP ARE CONJUGATE, Arxiv preprint arXiv:1108.1784, 2011
%H A192983 J. R. Britnell and M. Wildon, <a href="http://dx.doi.org/10.1515/JGT.2009.013">Commuting elements in conjugacy classes: an application of Hall's Marriage Theorem to group theory</a>, J. Group Theory, 12 (2009), 795-802.
%H A192983 Mark Wildon, <a href="http://www.ma.rhul.ac.uk/~uvah099/other.html#ConjugacyProbability">Haskell source code</a> for computing values of the sequence.
%e A192983 For n = 3 the probability that two elements of S_3 have conjugates that commute is a(3)/3!^2 = 2/3. Proof: only the transpositions and three cycles fail to have conjugates that commute; the probability of choosing one permutation from each of these classes is 2*1/2*1/3 = 1/3.
%o A192983 (Haskell) See links for code.
%Y A192983 Cf. A087132 (the sum of squares of the sizes of the conjugacy classes of S_n).
%K A192983 new,nonn
%O A192983 1,2
%A A192983 _Mark Wildon_, Aug 03 2011