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%I #29 Mar 17 2024 11:52:41
%S 1,384,49766400,2200248344641536,32699232783861202944000000,
%T 161655300770215803222365206216704000000,
%U 264237966861625003904099008804894577790426446838104064
%N Number of spanning trees in n X n X 2 grid.
%H W.-J. Tzeng and F. Y. Wu, <a href="https://arxiv.org/abs/cond-mat/0001408">Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces</a>, arXiv:cond-mat/0001408 [cond-mat.stat-mech], 2002.
%F a(n) = 2^(2*n^2-2) / n^2 * Product_{n1=0..n-1, n2=0..n-1, n3=0..1, n1+n2+n3>0} (3 - cos(Pi*n1/n) - cos(Pi*n2/n) - cos(Pi*n3/2)).
%F a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = (2*sqrt(2)-3)*(2+sqrt(3))*(sqrt(15)-4) = 0.08133113706589390743806107..., c = 5^(1/4) * Gamma(1/4) / (sqrt(3) * (2*Pi)^(3/4)) = 0.788729432659299631982768... and G is Catalan's constant A006752. Equivalently, m = Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx. - _Vaclav Kotesovec_, Jan 06 2021, updated Mar 17 2024
%t a[n_] := 2^(2*n^2 - 2)/n^2*Product[If[n1+n2+n3 > 0, 3 - Cos[Pi*n1/n] - Cos[Pi*n2/n] - Cos[Pi*n3/2], 1], {n1, 0, n-1}, {n2, 0, n-1}, {n3, 0, 1}];
%t Table[a[n] // Round, {n, 1, 7}] (* _Jean-François Alcover_, Feb 18 2019 *)
%Y Cf. A071763, A007341, A340396.
%K nonn
%O 1,2
%A Sharon Sela (sharonsela(AT)hotmail.com), Jun 08 2002
%E More terms from André Pönitz (poenitz(AT)htwm.de), Jun 11 2003
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