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
KEYWORD
nonn
AUTHOR
Drake Thomas, Jan 26 2021
EXTENSIONS
More terms from Seiichi Manyama, Feb 13 2021
STATUS
approved