A002899 Number of n-step polygons on f.c.c. lattice. (Formerly M4840 N2068)

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.

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

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

Cf. A002895, A002898.

Sequence in context: A326743 A269030 A222359 * A222966 A077612 A041272

Adjacent sequences: A002896 A002897 A002898 * A002900 A002901 A002902

Keyword

nonn,walk,nice

Author

N. J. A. Sloane

Extensions

More terms from David Broadhurst, Feb 06 2008

Status

approved