A067518 Number of spanning trees in n X n X 2 grid.

1, 384, 49766400, 2200248344641536, 32699232783861202944000000, 161655300770215803222365206216704000000, 264237966861625003904099008804894577790426446838104064

Offset

1,2

Links

Table of n, a(n) for n=1..7.

W.-J. Tzeng and F. Y. Wu, Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces, arXiv:cond-mat/0001408 [cond-mat.stat-mech], 2002.

Formula

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)).

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 = 0.08133113706589390743806107..., c = 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

Mathematica

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}];
Table[a[n] // Round, {n, 1, 7}] (* Jean-François Alcover, Feb 18 2019 *)

Crossrefs

Cf. A071763, A007341, A340396.

Sequence in context: A188783 A008692 A193313 * A071763 A254352 A274445

Adjacent sequences: A067515 A067516 A067517 * A067519 A067520 A067521

Keyword

nonn

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jun 08 2002

Extensions

More terms from André Pönitz (poenitz(AT)htwm.de), Jun 11 2003

Status

approved