A002899 Number of n-step polygons on f.c.c. lattice.

1, 0, 12, 48, 540, 4320, 42240, 403200, 4038300, 40958400, 423550512, 4434978240, 46982827584, 502437551616, 5417597053440, 58831951546368, 642874989479580, 7063600894137216, 77991775777488144, 864910651813116480

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• nice

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.
• Yen Lee Loh, A general method for calculating lattice Green functions on the branch cut, arXiv:1706.03083 [math-ph], 2017.
• Index entries for sequences related to f.c.c. lattice

Crossrefs

Cf. A002895, A002898.

Programs

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]

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

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

Extensions

More terms from David Broadhurst, Feb 06 2008