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A002899
Number of n-step polygons on f.c.c. lattice.
(Formerly M4840 N2068)
16
1, 0, 12, 48, 540, 4320, 42240, 403200, 4038300, 40958400, 423550512, 4434978240, 46982827584, 502437551616, 5417597053440, 58831951546368, 642874989479580, 7063600894137216, 77991775777488144, 864910651813116480
OFFSET
0,3
COMMENTS
a(n) is the number of 2 X n matrices with entries from {1,2,3,4}, with (1) second row a (multiset) permutation of the first, and (2) no constant columns. - David Callan, Aug 25 2009
a(n) is the constant coefficient in the expansion of (x + y + z + 1/x + 1/y + 1/z + x/y + y/z + z/x + y/x + z/y + x/z)^n. - Seiichi Manyama, Oct 26 2019
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Christoph Koutschan, Table of n, a(n) for n = 0..931
David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
FORMULA
G.f.: hypergeom([1/6, 1/3],[1],108*x^2*(4*x+1))^2. - Mark van Hoeij, Oct 29 2011
Recurrence: n^3*a(n) - 2*n*(2*n-1)*(n-1)*a(n-1) - 16*(n-1)*(5*n^2-10*n+6)*a(n-2) - 96*(n-1)*(n-2)*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Dec 10 2013
a(n) ~ 2^(2*n-2) * 3^(n+3/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 08 2016
MATHEMATICA
f[n_] := Sum[ Binomial[n, k]*(-4)^(n - k)*Sum[ Binomial[k, j]^2*Binomial[2k - 2j, k - j]*Binomial[2j, j], {j, 0, k}], {k, 0, n}]; Array[f, 20, 0]
PROG
(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(-4)^(n-k)*sum(j=0, k, binomial(k, j)^2*binomial(2*k-2*j, k-j)*binomial(2*j, j)))};
print(vector(20, n, a(n-1))) \\ David Broadhurst, Feb 06 2008; fixed by Vaclav Kotesovec, Apr 08 2016
CROSSREFS
Sequence in context: A326743 A269030 A222359 * A222966 A376185 A077612
KEYWORD
nonn,walk,nice
EXTENSIONS
More terms from David Broadhurst, Feb 06 2008
STATUS
approved