Search: 2992
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A006331
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a(n) = n*(n+1)*(2n+1)/3.
(Formerly M1963)
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+20
34
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0, 2, 10, 28, 60, 110, 182, 280, 408, 570, 770, 1012, 1300, 1638, 2030, 2480, 2992, 3570, 4218, 4940, 5740, 6622, 7590, 8648, 9800, 11050, 12402, 13860, 15428, 17110, 18910, 20832, 22880, 25058, 27370, 29820, 32412, 35150, 38038, 41080, 44280
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OFFSET
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0,2
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COMMENTS
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Triangles in rhombic matchstick arrangement of side n.
Maximum accumulated number of electrons at energy level n. - Scott A. Brown (scottbrown(AT)neo.rr.com), Feb 28 2000
Let M_n denote the n X n matrix M_n(i,j)=i^2+j^2; then the characteristic polynomial of M_n is x^n - a(n)x^(n-1) - .... - Michael Somos, Nov 14 2002
Convolution of odds (A005408) and evens (A005843). - Graeme McRae, Jun 06 2006
10*a(n) = A016755(n) - A001845(n); since A016755 are odd cubes and A001845 centered octahedral numbers, 10*a(n) are the "odd cubes without their octahedral contents." - Damien Pras, Mar 19 2011
a(n) is the number of non-monotonic functions with domain {0,1,2} and codomain {0,1,...,n}. - Dennis P. Walsh, Apr 25 2011
For any odd number 2n+1, find sum a*b, {a<b and a+b=2n+1}. This sum is equal to the n-th nonzero term of this sequence. Thus for 13=2*n+1, n=6; there are six products 1*12+2*11+3*10+4*9+5*8+6*7=182, which is also twice the sum of the squares for n=6. - J. M. Bergot, Jul 16 2011
a(n) gives the number of (n+1) X (n+1) symmetric (0,1)-matrices containing three ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and |w-x|<y. - Clark Kimberling, Jun 02 2012
Partial sums of A001105. - Omar E. Pol, Jan 12 2013
Total number of square diagonals (of any size) in an n X n square grid. - Wesley Ivan Hurt, Mar 24 2015
Number of diagonal attacks of two queens on (n+1) X (n+1) chessboard. - Antal Pinter, Sep 20 2015
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..10000
J. L. Bailey, Jr., A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359.
J. L. Bailey, A table to facilitate the fitting of certain logistic curves, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
G. Kreweras, Sur une classe de problèmes de denombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle}, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
Author?, Basic atomic information [broken link ?]
Dennis Walsh, Notes on finite monotonic and non-monotonic functions
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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G.f.: 2*x*(1+x)/(1-x)^4. - Simon Plouffe (in his 1992 dissertation)
a(n) = 2*binomial(n+1,3) + 2*binomial(n+2,3).
a(n) = 2*A000330(n) = A002492(n)/2.
From the formula for the sum of squares of positive integers 1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2*n+1)/6, if we multiply both sides by 2 we get Sum_{k=0..n} 2*k^2 = n(n+1)(2*n+1)/3, which is an alternative formula for this sequence. - Mike Warburton (mikewarb(AT)gmail.com), Sep 08 2007
a(n) = sum(a*b), where the summing is over all unordered partitions 2*n+1=a+b. - Vladimir Shevelev, May 11 2012
a(n) = binomial(2n+2, 3)/2. - Ronan Flatley, Dec 13 2012
a(n) = A000292(n) + A002411(n). - Omar E. Pol, Jan 11 2013
a(0)=0, a(1)=2, a(2)=10, a(3)=28, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Apr 12 2013
a(n) = A208532(n+1,2). - Philippe Deléham, Dec 05 2013
Sum_{n>0} 1/a(n) = 9 - 12*log(2). - Enrique Pérez Herrero, Dec 03 2014
a(n) = A000292(n-1) + (n+1)*A000217(n). - J. M. Bergot, Sep 02 2015
a(n) = 2*(A000332(n+3) - A000332(n+1)). - Antal Pinter, Sep 20 2015
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EXAMPLE
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For n=2, a(2)=10 since there are 10 non-monotonic functions f from {0,1,2} to {0,1,2}, namely, functions f = <f(1),f(2),f(3)> given by <0,1,0>, <0,2,0>, <0,2,1>, <1,0,1>, <1,0,2>, <1,2,0>, <1,2,1>, <2,0,1>, <2,0,2>, and <2,1,2>. - Dennis P. Walsh, Apr 25 2011
Let n=4, 2*n+1 = 9. Since 9 = 1+8 = 3+6 = 5+4 = 7+2, a(4) = 1*8 + 3*6 + 5*4 + 7*2 = 60. - Vladimir Shevelev, May 11 2012
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MAPLE
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A006331 := proc(n)
n*(n+1)*(2*n+1)/3 ;
end proc:
seq(A006331(n), n=0..80) ; # R. J. Mathar, Sep 27 2013
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MATHEMATICA
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Table[n(n+1)(2n+1)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 2, 10, 28}, 50] (* Harvey P. Dale, Apr 12 2013 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n*(n+1)*(2*n+1)/3)
(MAGMA) [n*(n+1)*(2*n+1)/3: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
(Haskell)
a006331 n = sum $ zipWith (*) [2*n-1, 2*n-3 .. 1] [2, 4 ..]
-- Reinhard Zumkeller, Feb 11 2012
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CROSSREFS
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A row of A132339.
a(n) = Sum_{i=0..n} T(i,n-i), array T as in A048147.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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A033001
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Every run of digits of n in base 3 has length 2.
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+20
31
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4, 8, 36, 44, 72, 76, 328, 332, 396, 400, 652, 656, 684, 692, 2952, 2960, 2988, 2992, 3568, 3572, 3600, 3608, 5868, 5876, 5904, 5908, 6160, 6164, 6228, 6232, 26572, 26576, 26640, 26644, 26896, 26900, 26928, 26936, 32112, 32120
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OFFSET
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1,1
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COMMENTS
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See A043291 for the base 2 version (which has a very simple formula), A033002 - A033014 for bases 4 through 16, A033015 - A033029 for the variants with runs of length >= 2. - M. F. Hasler, Feb 01 2014
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..250
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FORMULA
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a(n)=4*A043307(n). - M. F. Hasler, Feb 01 2014
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MATHEMATICA
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Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 3]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)
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PROG
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From M. F. Hasler, Feb 01 2014 (Start)
(PARI) is_A033001(n)=!until(!n\=9, bittest(4588304, n%27)||return)
(PARI) for(n=1, 9999, is_A033001(n)&&print1(n", ")) \\ (End)
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KEYWORD
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nonn,base
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AUTHOR
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Clark Kimberling
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STATUS
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approved
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A005993
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G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2).
(Formerly M1576)
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+20
30
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1, 2, 6, 10, 19, 28, 44, 60, 85, 110, 146, 182, 231, 280, 344, 408, 489, 570, 670, 770, 891, 1012, 1156, 1300, 1469, 1638, 1834, 2030, 2255, 2480, 2736, 2992, 3281, 3570, 3894, 4218, 4579, 4940, 5340, 5740, 6181, 6622, 7106, 7590, 8119, 8648, 9224, 9800
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OFFSET
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0,2
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COMMENTS
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Alkane (or paraffin) numbers l(6,n).
Dimension of the space of homogeneous degree n polynomials in (x1, y1, x2, y2) invariant under permutation of variables x1<->y1, x2<->y2.
Also multidigraphs with loops on 2 nodes with n arcs (see A136564). - Vladeta Jovovic, Dec 27 1999
Euler transform of finite sequence [2,3,0,-1]. - Michael Somos, Mar 17 2004
a(n-2) is the number of plane partitions with trace 2. - Michael Somos, Mar 17 2004
With offset 4, a(n) is the number of bracelets with n beads, 3 of which are red, 1 of which is blue. For odd n, a(n) = C(n-1,3)/2. For even n, a(n) = C(n-1,3)/2 +(n-2)/4. For n >= 6, with K = (n-1)(n-2)/((n-5)(n-4)), for odd n, a(n) = K*a(n-2). For even n, a(n) = K*a(n-2) -(n-2)/(n-5). - Washington Bomfim, Aug 05 2008
Equals (1,2,3,4,...) convolved with (1,0,3,0,5,...). - Gary W. Adamson, Feb 16 2009
Equals row sums of triangle A177878.
Equals (1/2)*((1, 4, 10, 20, 35, 56,...) + (1, 0, 2 0, 3, 0, 4,...)).
Comment from Ctibor O. Zizka, Nov 21 2014 (Start)
With offset 4, a(n) is the number of different patterns of the 2-color 4-partition of n.
P(n)_(k;t) counts the number of different patterns of the t-color, k-partition of n.
P(n)_(k;t) = 1 + Sum(i=2..n)Sum(j=2..i)Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i).
P(n;i;j) = Sum(r=1..m) c_(i,j)*v_r*F_r(X_1,...,X_i).
m partition number of i.
c_(i,j) number of different coloring patterns on the r-th form (X_1,...,X_i) of i-partition with j-colors.
v_r number of i-partitions of n of the r-th form (X_1,...,X_i).
F_r(X_1,...,X_i) number of different patterns of the r-th form i-partition of n.
Some simple results:
P(1)_(k;t)=1 , P(2)_(k;t)=2 , P(3)_(k;t)=4 , P(4)_(k;t)=11 etc.
P(n;1;1) = P(n;n;n) = 1 for all n;
P(n;2;2) = floor(n/2) (A004526);
P(n;3;2) = (n*n - 2*n + n mod 2)/4 (A002620).
This sequence is a(n) = P(n;4;2).
2-coloring of 4-partition is (A,B,A,B) or (B,A,B,A).
Each 4-partition of n has one of the form (X_1,X_1,X_1,X_1),(X_1,X_1,X_1,X_2), (X_1,X_1,X_2,X_2),(X_1,X_1,X_2,X_3),(X_1,X_2,X_3,X_4).
The number of forms is m=5 which is the partition number of k=4.
Partition form (X_1,X_1,X_1,X_1) gives 1 pattern ((X_1A,X_1B,X_1A,X_1B),
(X_1,X_1,X_1,X_2) gives 2 patterns,(X_1,X_1,X_2,X_2) gives 4 patterns,(X_1,X_1,X_2,X_3) gives 6 patterns and (X_1,X_2,X_3,X_4) gives 12 patterns.
Thus
a(n) = P(n;4;2) = 1*1*v_1 + 1*2*v_2 + 1*4*v_3 + 1*6*v_4 + 1*12*v_5
where v_r is the number of different 4-partitions of the r-th form (X_1,X_2,X_3,X_4) for a given n.
Example:
The 4-partitions of 8 are (2,2,2,2), (1,1,1,5), (1,1,3,3), (1,1,2,4), and (1,2,2,3
(2,2,2,2) 1 pattern
(1,1,1,5), (1,1,5,1) 2 patterns
(1,1,3,3), (1,3,3,1), (3,1,1,3), (1,3,1,3) 4 patterns
(1,1,2,4), (1,1,4,2), (1,2,1,4), (1,2,4,1), (1,4,1,2), (2,1,1,4) 6 patterns
(2,2,1,3), (2,2,3,1), (2,1,2,3), (2,1,3,2), (2,3,2,1), (1,2,2,3) 6 patterns
Thus a(8) = P(8,4,2) = 1+2+4+6+6 = 19.
(End)
a(n) = length of run n+2 of consecutive 1's in A254338. - Reinhard Zumkeller, Feb 27 2015
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 96.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
M. Benoumhani, M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 3rd ine.
Washington Bomfim, The 19 bracelets with 8 beads - one blue, three reds and four blacks. [From Washington Bomfim, Aug 05 2008]
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 8.
Naihuan Jing, Kailash Misra, Carla Savage, On multi-color partitions and the generalized Rogers-Ramanujan identities, arXiv:math/9907183 [math.CO], 1999.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane, Classic Sequences
L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250. See page 218. MR1433171 (98i:13009).
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
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FORMULA
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l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
G.f.: (1+x^2)/((1-x)^2*(1-x^2)^2) = (1+x^2)/((1+x)^2*(x-1)^4) = (1/(1-x)^4 +1/(1-x^2)^2)/2.
a(2n) = (n+1)(2n^2+4n+3)/3, a(2n+1) = (n+1)(n+2)(2n+3)/3. a(-4-n) = -a(n).
From Yosu Yurramendi, Sep 12 2008: (Start)
a(n+1) = a(n) + A008794(n+3) with a(1)=1.
a(n) = A027656(n) + 2*A006918(n).
a(n+2) = a(n) + A000982(n+2) with a(1)=1, a(2)=2. (End)
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Jaume Oliver Lafont, Dec 05 2008
a(n) = (n^3 + 6*n^2 + 11*n + 6)/12 + ((n+2)/4)[n even] (the bracket means that the second term is added if and only if n is even). - Benoit Jubin, Mar 31 2012
a(n) = (1/12)*n*(n+1)*(n+2) + (1/4)*(n+1)*(1/2)*(1-(-1)^n), with offset 1. - Yosu Yurramendi, Jun 20 2013
a(n) = Sum_{i=0..n+1} ceiling(i/2) * round(i/2) = Sum_{i=0..n+2} floor(i/2)^2. - Bruno Berselli, Aug 30 2013
a(n) = (n + 2)*(3*(-1)^n + 2*n^2 + 8*n + 9)/24. - Ilya Gutkovskiy, May 04 2016
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EXAMPLE
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a(2) = 6, since ( x1*y1, x2*y2, x1*x1+y1*y1, x2*x2+y2*y2, x1*x2+y1*y2, x1*y2+x2*y1 ) are a basis for homogeneous quadratic invariant polynomials.
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MAPLE
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g := proc(n) local i; add(floor(i/2)^2, i=1..n+1) end: # Joseph S. Riel (joer(AT)k-online.com), Mar 22 2002
a:= n-> (Matrix([[1, 0$3, -1, -2]]).Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..44); # Alois P. Heinz, Jul 31 2008
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MATHEMATICA
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CoefficientList[Series[(1+x^2)/((1-x)^2*(1-x^2)^2), {x, 0, 44}], x] (* Jean-François Alcover, Apr 08 2011 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 2, 6, 10, 19, 28}, 50] (* Harvey P. Dale, Feb 20 2012 *)
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PROG
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(PARI) a(n)=polcoeff((1+x^2)/(1-x)^2/(1-x^2)^2+x*O(x^n), n)
(PARI) a(n) = (binomial(n+3, n) + (1-n%2)*binomial((n+2)/2, n>>1))/2. \\ Washington Bomfim, Aug 05 2008]
(Haskell) Following Gary W. Adamson.
import Data.List (inits, intersperse)
a005993 n = a005994_list !! n
a005993_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
tail $ inits [1..]
-- Reinhard Zumkeller, Feb 27 2015
(MAGMA) I:=[1, 2, 6, 10, 19, 28]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-2)-4*Self(n-3)+Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 19 2015
(Sage)
def A005993():
a, b, to_be = 0, 0, True
while True:
yield (a*(a*(2*a+9)+13)+b*(b+1)*(2*b+1)+6)//6
if to_be: b += 1
else: a += 1
to_be = not to_be
a = A005993()
[a.next() for _ in range(48)] # Peter Luschny, May 04 2016
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CROSSREFS
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Cf. A177878.
Partial sums of A008794 (without 0). - Bruno Berselli, Aug 30 2013
Cf. A254338, A002260, A005408.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)
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STATUS
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approved
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A179644
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Product of the 4th power of a prime and 2 different distinct primes (p^4*q*r).
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+20
26
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240, 336, 528, 560, 624, 810, 816, 880, 912, 1040, 1104, 1134, 1232, 1360, 1392, 1456, 1488, 1520, 1776, 1782, 1840, 1904, 1968, 2064, 2106, 2128, 2256, 2288, 2320, 2480, 2544, 2576, 2754, 2832, 2835, 2928, 2960, 2992, 3078, 3216, 3248, 3280, 3344, 3408
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OFFSET
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1,1
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COMMENTS
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240=2^4*3*5,336=2^4*3*7,..810=2^3^4*5,..
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Prime Signatures
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MATHEMATICA
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f[n_]:=Sort[Last/@FactorInteger[n]]=={1, 1, 4}; Select[Range[4000], f]
Take[Union[#[[1]]^4 #[[2]]#[[3]]&/@(Flatten[Permutations/@ Subsets[ Prime[ Range[ 20]], {3}], 1])], 50] (* Harvey P. Dale, Feb 07 2013 *)
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PROG
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(PARI) list(lim)=my(v=List(), t); forprime(p=2, (lim\6)^(1/4), forprime(q=2, sqrt(lim\p^4), if(p==q, next); t=p^4*q; forprime(r=q+1, lim\t, if(p==r, next); listput(v, t*r)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 19 2011
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CROSSREFS
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Cf. A006881, A007304, A065036, A085986, A085987, A178739, A179642, A179643.
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KEYWORD
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nonn
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AUTHOR
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Vladimir Joseph Stephan Orlovsky, Jul 21 2010
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STATUS
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approved
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A135302
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Square array of numbers A(n,k) (n>=0, k>=0) of transitive reflexive early confluent binary relations R on n labeled elements where |{y : xRy}| <= k for all x, read by antidiagonals.
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+20
20
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 13, 4, 1, 1, 0, 1, 62, 26, 4, 1, 1, 0, 1, 311, 168, 26, 4, 1, 1, 0, 1, 1822, 1416, 243, 26, 4, 1, 1, 0, 1, 11593, 13897, 2451, 243, 26, 4, 1, 1, 0, 1, 80964, 153126, 29922, 2992, 243, 26, 4, 1, 1, 0, 1, 608833, 1893180, 420841
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OFFSET
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0,13
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COMMENTS
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R is early confluent iff (xRy and xRz) implies (yRz or zRy) for all x, y, z.
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REFERENCES
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A. P. Heinz (1990). Analyse der Grenzen und Möglichkeiten schneller Tableauoptimierung. PhD Thesis, Albert-Ludwigs-Universität Freiburg, Freiburg i. Br., Germany.
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LINKS
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Alois P. Heinz, Antidiagonals n = 0..55, flattened
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FORMULA
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E.g.f. of column k=0: t_0(x) = 1; e.g.f. of column k>0: t_k(x) = exp (Sum_{m=1..k} x^m/m! * t_{k-m}(x)).
A(n,k) = Sum{i=0..k} A135313(n,i).
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EXAMPLE
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Table A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, ...
0, 1, 4, 4, 4, 4, ...
0, 1, 13, 26, 26, 26, ...
0, 1, 62, 168, 243, 243, ...
0, 1, 311, 1416, 2451, 2992, ...
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MAPLE
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t:= proc(k) option remember; `if`(k<0, 0,
unapply(exp(add(x^m/m! *t(k-m)(x), m=1..k)), x))
end:
A:= proc(n, k) option remember;
coeff(series(t(k)(x), x, n+1), x, n) *n!
end:
seq(seq(A(d-i, i), i=0..d), d=0..15);
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MATHEMATICA
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t[0, _] = 1; t[k_, x_] := t[k, x] = Exp[Sum[x^m/m!*t[k-m, x], {m, 1, k}]]; a[0, 0] = 1; a[_, 0] = 0; a[n_, k_] := SeriesCoefficient[t[k, x], {x, 0, n}]*n!; Table[a[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2013, after Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000012, A135312, A210911, A210912, A210913, A210914, A210915, A210916, A210917, A210918.
Main diagonal gives A052880.
A(n,n)-A(n,n-1) gives A000670.
Cf. A135313.
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KEYWORD
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nonn,tabl
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AUTHOR
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Alois P. Heinz, Dec 04 2007
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STATUS
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approved
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A085939
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Horadam sequence (0,1,6,4).
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+20
15
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0, 1, 4, 22, 112, 580, 2992, 15448, 79744, 411664, 2125120, 10970464, 56632576, 292353088, 1509207808, 7790949760, 40219045888, 207621882112, 1071801803776, 5532938507776, 28562564853760
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OFFSET
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0,3
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COMMENTS
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a(n) / a(n-1) converges to 10^1/2 + 2 as n approaches infinity. 10^1/2 + 2 can also be written as 2^1/2 * (2^1/2 + 5^1/2), ((2 * 2^1/2) * Phi) - 2^1/2 + 2 and 2^1/2 * (2^1/2 + (L(n) / F(n))), where L(n) is the n-th Lucas number and F(n) is the n-th Fibonacci number as n approaches infinity.
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LINKS
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Table of n, a(n) for n=0..20.
Eric Weisstein, Lucas Number
Eric Weisstein, Lucas Sequence
Eric Weisstein, Horadam Sequence
Eric Weisstein, Fibonacci Number
Eric Weisstein, Pell Number
Index entries for linear recurrences with constant coefficients, signature (4, 6).
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FORMULA
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a(n) = s*a(n-1) + r*a(n-2); for n > 1, where a(0) = 0, a(1) = 1, s = 4, r = 6
a(n)=((2+sqrt(10))^n-(2-sqrt(10))^n)/(2*sqrt(10)) [From Rolf Pleisch, Jul 06 2009]
G.f.: x/(1-4*x-6*x^2). [Colin Barker, Jan 10 2012]
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EXAMPLE
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a(4) = 112 because a(3) = 22, a(2) = 4, s = 4, r = 6 and (4 * 22) + (6 * 4) = 112.
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MATHEMATICA
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Join[{a=0, b=1}, Table[c=4*b+6*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
LinearRecurrence[{4, 6}, {0, 1}, 30] (* Harvey P. Dale, Jul 20 2016 *)
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PROG
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(Sage) [lucas_number1(n, 4, -6) for n in xrange(0, 21)]# [From Zerinvary Lajos, Apr 23 2009]
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CROSSREFS
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Cf. A024318, A000032, A000129.
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KEYWORD
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easy,nonn
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AUTHOR
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Ross La Haye, Aug 16 2003
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STATUS
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approved
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A253961
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T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with every 3X3 subblock diagonal median minus antidiagonal median nondecreasing horizontally, vertically and ne-to-sw antidiagonally
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+20
14
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512, 2992, 2992, 16544, 20868, 16544, 82064, 142800, 148332, 82064, 423232, 1022752, 1501724, 1048679, 423232, 2154496, 7533252, 15946360, 16186768, 7777096, 2154496, 11244672, 55225242, 167174472, 240903042, 168988280
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OFFSET
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1,1
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COMMENTS
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Table starts
......512......2992.......16544........82064.........423232..........2154496
.....2992.....20868......142800......1022752........7533252.........55225242
....16544....148332.....1501724.....15946360......167174472.......1776422573
....82064...1048679....16186768....240903042.....3635332193......55074999330
...423232...7777096...168988280...3624943928....77993064012....1716319825801
..2154496..56216811..1798654745..55018824334..1717613401125...54401550145686
.11244672.413408572.18976071008.841599218625.37660244542860.1720060910899929
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LINKS
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R. H. Hardin, Table of n, a(n) for n = 1..283
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FORMULA
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Empirical for column k:
k=1: [linear recurrence of order 8] for n>15
Empirical for row n:
n=1: [same linear recurrence of order 8] for n>15
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EXAMPLE
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Some solutions for n=2 k=4
..0..0..0..1..0..1....0..0..0..0..0..0....0..0..0..0..0..0....0..0..0..1..1..1
..1..0..1..1..1..1....0..1..1..0..1..0....1..0..0..0..0..0....0..1..1..0..0..0
..1..0..1..1..1..0....0..0..0..0..1..0....1..0..1..0..1..1....1..0..0..1..0..1
..0..0..0..1..0..0....0..0..0..0..1..1....0..0..0..0..1..0....0..1..1..0..1..0
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CROSSREFS
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Column 1 and row 1 are A253757
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KEYWORD
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nonn,tabl
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AUTHOR
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R. H. Hardin, Jan 20 2015
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STATUS
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approved
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A031173
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Longest edge a of smallest (measured by the longest edge) primitive Euler bricks (a, b, c, sqrt(a^2 + b^2), sqrt(b^2 + c^2), sqrt(a^2 + c^2) are integers).
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+20
12
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240, 275, 693, 720, 792, 1155, 1584, 2340, 2640, 2992, 3120, 5984, 6325, 6336, 6688, 6732, 8160, 9120, 9405, 10725, 11220, 12075, 13860, 14560, 16800, 17472, 17748, 18560, 19305, 21476, 23760, 23760, 24684, 25704, 26649, 29920, 30780
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OFFSET
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1,1
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COMMENTS
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Primitive means that GCD(a,b,c) = 1.
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REFERENCES
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RR Gallyamov, IR Kadyrov, DD Kashelevskiy, A fast modulo primes algorithm for searching perfect cuboids and its implementation, arXiv preprint arXiv:1601.00636, 2016
Calculated by F. Helenius (fredh(AT)ix.netcom.com).
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LINKS
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Giovanni Resta, Table of n, a(n) for n = 1..3556
A. A. Masharov and R. A. Sharipov, A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture, arXiv preprint, 2015.
Giovanni Resta, The 3556 primitive bricks with c < b < a < 5*10^8
J. Ramsden and H. Sharipov, Inverse problems associated with perfect cuboids, arXiv preprint arXiv:1207.6764, 2012. - From N. J. A. Sloane, Dec 23 2012
J. Ramsden and H. Sharipov, On singularities of the inverse problems associated with perfect cuboids, arXiv preprint arXiv:1208.1859, 2012. - From N. J. A. Sloane, Dec 25 2012
Ruslan Sharipov, Perfect cuboids and irreducible polynomials, arXiv:1108.5348, 2011
Ruslan Sharipov, A note on the first cuboid conjecture, arXiv:1109.2534, 2011
Ruslan Sharipov, A note on the second cuboid conjecture. Part I, arXiv:1201.1229, 2012
Ruslan Sharipov, Perfect cuboids and multisymmetric polynomials, arXiv preprint arXiv:1205.3135, 2012. - From N. J. A. Sloane, Oct 22 2012
Ruslan Sharipov, On an ideal of multisymmetric polynomials associated with perfect cuboids, arXiv preprint arXiv:1206.6769, 2012. - From N. J. A. Sloane, Dec 17 2012
Ruslan Sharipov, On the equivalence of cuboid equations and their factor equations, arXiv preprint arXiv:1207.2102, 2012. - From N. J. A. Sloane, Dec 22 2012
Ruslan Sharipov, A biquadratic Diophantine equation associated with perfect cuboids, arXiv preprint arXiv:1207.4081, 2012. - From N. J. A. Sloane, Dec 23 2012
Ruslan Sharipov, On a pair of cubic equations associated with perfect cuboids, arXiv preprint arXiv:1208.0308, 2012. - From N. J. A. Sloane, Dec 23 2012
Ruslan Sharipov, On two elliptic curves associated with perfect cuboids, arXiv preprint arXiv:1208.1227, 2012. - From N. J. A. Sloane, Dec 24 2012
Ruslan Sharipov, Asymptotic estimates for roots of the cuboid characteristic equation in the linear region, arXiv preprint, 2015.
Ruslan Sharipov, Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture, arXiv preprint, 2015.
Ruslan Sharipov, A note on invertible quadratic transformations of the real plane, arXiv preprint arXiv:1507.01861, 2015
Eric Weisstein's World of Mathematics, Euler Brick
Index entries for sequences related to bricks
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CROSSREFS
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Cf. A031174, A031175.
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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A023855
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a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + ... + (n+1-k)*k, where k = floor((n+1)/2).
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+20
11
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1, 2, 7, 10, 22, 28, 50, 60, 95, 110, 161, 182, 252, 280, 372, 408, 525, 570, 715, 770, 946, 1012, 1222, 1300, 1547, 1638, 1925, 2030, 2360, 2480, 2856, 2992, 3417, 3570, 4047, 4218, 4750, 4940, 5530, 5740, 6391, 6622, 7337, 7590, 8372, 8648, 9500, 9800, 10725, 11050
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OFFSET
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1,2
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COMMENTS
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Given a rectangle of perimeter 2*n one can form rectangles having this perimeter for a number of different rectangles or squares depending on how large 2*n is. The sequence lists the total areas of all such rectangles for each 2*n. [J. M. Bergot, Sep 14 2011]
Conjecture: Antidiagonal sums of triangle A075462. - L. Edson Jeffery, Jan 20 2012
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1)
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FORMULA
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a(n) = (n+1)*(n+3)*(2*n+1)/24 if n is odd, or n*(n+1)*(n+2)/12 if n is even
G.f.: x*(1+x+2*x^2)]/((1-x)^4*(1+x)^3). - Ralf Stephan, Apr 28 2004
a(n) = sum_{i=1..ceil(n/2)} i*(n-i+1) = -ceil(n/2)*(ceil(n/2)+1)*(2*ceil(n/2)-3n-2)/6. - Wesley Ivan Hurt, Sep 19 2013
a(n)= (4*n^3+15*n^2+14*n+3-3*(n+1)^2*(-1)^n)/48. - Luce ETIENNE, Oct 22 2014
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MAPLE
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seq(-(1/3)*floor((k+1)/2)^3 + (k/2)*floor((k+1)/2)^2 + ((3*k+2)/6)*floor((k+1)/2), k=1..100); # Wesley Ivan Hurt, Sep 18 2013
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MATHEMATICA
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LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 2, 7, 10, 22, 28, 50}, 50] (* Vincenzo Librandi, Jan 23 2012 *)
Table[-Ceiling[n/2] (Ceiling[n/2] + 1) (2 Ceiling[n/2] - 3 n - 2)/6, {n, 100}] (* Wesley Ivan Hurt, Sep 20 2013 *)
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PROG
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(PARI) a(n)=if(n%2, (n+1)*(n+3)*(2*n+1)/24, n*(n+1)*(n+2)/12)
(Haskell)
a023855 n = sum $ zipWith (*) [1 .. div (n+1) 2] [n, n-1 ..]
-- Reinhard Zumkeller, Jan 23 2012
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CROSSREFS
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Cf. A023856, A023857, A024305, A024854.
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KEYWORD
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nonn,easy
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Formula, program, and slight revision by Charles R Greathouse IV, Feb 23 2010
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STATUS
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approved
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A056524
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Palindromes with even number of digits.
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+20
11
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11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554
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OFFSET
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1,1
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COMMENTS
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Concatenation of n with reverse of n (keeping leading zeros in the reverse).
A178788(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Jun 30 2010
All of the terms are divisible by eleven. - James Burling, Aug 08 2014
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = n*10^A055642(n) + A004086(n).
a(n) = 11 * A066492(n).
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MATHEMATICA
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d[n_]:=IntegerDigits[n]; Table[FromDigits[Join[x=d[n], Reverse[x]]], {n, 45}] (* Jayanta Basu, May 29 2013 *)
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PROG
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(Haskell)
a056524 n = a056524_list !! (n-1)
a056524_list = [read (ns ++ reverse ns) :: Integer |
n <- [0..], let ns = show n]
-- Reinhard Zumkeller, Dec 28 2011
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CROSSREFS
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Cf. A002113, A004086, A056525, A066492.
Cf. A020338.
Cf. A110745 (permutation).
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KEYWORD
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base,easy,nonn
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AUTHOR
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Henry Bottomley, Jun 16 2000
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STATUS
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approved
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