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%I M4840 N2068 #54 Jan 31 2022 06:47:12
%S 1,0,12,48,540,4320,42240,403200,4038300,40958400,423550512,
%T 4434978240,46982827584,502437551616,5417597053440,58831951546368,
%U 642874989479580,7063600894137216,77991775777488144,864910651813116480
%N Number of n-step polygons on f.c.c. lattice.
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Christoph Koutschan, <a href="/A002899/b002899.txt">Table of n, a(n) for n = 0..931</a>
%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008.
%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.
%H Yen Lee Loh, <a href="https://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.
%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%F G.f.: hypergeom([1/6, 1/3],[1],108*x^2*(4*x+1))^2. - _Mark van Hoeij_, Oct 29 2011
%F 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
%F a(n) ~ 2^(2*n-2) * 3^(n+3/2) / (Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Apr 08 2016
%t 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]
%o (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)))};
%o print(vector(20, n, a(n-1))) \\ _David Broadhurst_, Feb 06 2008; fixed by _Vaclav Kotesovec_, Apr 08 2016
%Y Cf. A002895, A002898.
%K nonn,walk,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _David Broadhurst_, Feb 06 2008