A335586 Number of domino tilings of a 2n X 2n toroidal grid.

1, 8, 272, 90176, 311853312, 11203604497408, 4161957566985310208, 15954943354032349049274368, 630665326543010382995142219988992, 256955886436135671144699761794930161483776

Offset

0,2

Comments

For n > 1, number of perfect matchings of the graph C_2n X C_2n.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..44

S. N. Perepechko, The number of perfect matchings on C_m X C_n graphs, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.

Drake Thomas, 8 tilings for the 2 X 2 toroidal grid.

Eric Weisstein's World of Mathematics, Perfect Matching

Eric Weisstein's World of Mathematics, Torus Grid Graph

Formula

a(n) = 4 * Product_{j=1..n-1} Product_{k=1..n} (4*sin(j*Pi/n)^2 + 4*sin((2*k-1)*Pi/(2*n))^2) + 1/2 * Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/(2*n))^2) = 4 * A341478(n)^2 + A341479(n)/2 for n > 0. - Seiichi Manyama, Feb 13 2021

a(n) ~ (1 + sqrt(2)) * exp(4*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

Example

For n = 1, there are a(1) = 8 tilings (see the Links section for a diagram).

Prog

(PARI) default(realprecision, 120);
b(n) = round(prod(j=1, n-1, prod(k=1, n, 4*sin(j*Pi/n)^2+4*sin((2*k-1)*Pi/(2*n))^2)));
c(n) = round(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/(2*n))^2)));
a(n) = if(n==0, 1, 4*b(n)+c(n)/2); \\ Seiichi Manyama, Feb 13 2021

Crossrefs

Number of perfect matchings of the graph C_2m X C_n: A162484 (m=1), A220864 (m=2), A232804 (m=3), A253678 (m=4), A281679 (m=5), A309018 (m=6).

Cf. A004003, A212800, A340562, A341478, A341479.

Sequence in context: A129424 A274559 A338636 * A159496 A045479 A179570

Adjacent sequences: A335583 A335584 A335585 * A335587 A335588 A335589

Keyword

nonn

Author

Drake Thomas, Jan 26 2021

Extensions

More terms from Seiichi Manyama, Feb 13 2021

Status

approved