A071763 Number of spanning trees in n X n X n grid.

1, 384, 8193540096000, 172685928902844729688524604506636288, 77746347057132811936046563068332100246216273086593103906734080000000000000

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jun 04 2002

Keywords

• nonn

Links

• W.-J. Tzeng and F. Y. Wu, Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces, arXiv:cond-mat/0001408 [cond-mat.stat-mech], 2000.
• Eric Weisstein's World of Mathematics, Grid Graph
• Eric Weisstein's World of Mathematics, Spanning Tree

Crossrefs

Cf. A007341, A340182.

Programs

Mathematica

Table[2^(n^3 - 1)/n^3 Product[Piecewise[{{1, i == j == k == 0}}, 3 - Cos[Pi i/n] - Cos[Pi j/n] - Cos[Pi k/n]], {i, 0, n - 1}, {j, 0, n - 1}, {k, 0, n - 1}], {n, 12}] // Round

Formula

a(n) = 2^(n^3-1) / n^3 * Product_{n1=0..n-1 n2=0..n-1 n3=0..n-1} (3- cos(Pi*n1/n) - cos(Pi*n2/n) - cos(Pi*n3/n) ) where n1, n2, n3 are not all 0.

Limit_{n->infinity} a(n)^(1/n^3) = exp(8 * A340322 / Pi^3) = 5.330202889205167421134597996649659520108446730592285502966091902480522584119... - Vaclav Kotesovec, Jan 05 2021