cp's OEIS Frontend

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

%I A335586 #60 Feb 16 2025 08:34:00
%S A335586 1,8,272,90176,311853312,11203604497408,4161957566985310208,
%T A335586 15954943354032349049274368,630665326543010382995142219988992,
%U A335586 256955886436135671144699761794930161483776
%N A335586 Number of domino tilings of a 2n X 2n toroidal grid.
%C A335586 For n > 1, number of perfect matchings of the graph C_2n X C_2n.
%H A335586 Seiichi Manyama, <a href="/A335586/b335586.txt">Table of n, a(n) for n = 0..44</a>
%H A335586 S. N. Perepechko, <a href="http://www.jip.ru/2016/333-361-2016.pdf">The number of perfect matchings on C_m X C_n graphs</a>, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.
%H A335586 Drake Thomas, <a href="/A335586/a335586.png">8 tilings for the 2 X 2 toroidal grid</a>.
%H A335586 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectMatching.html">Perfect Matching</a>
%H A335586 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TorusGridGraph.html">Torus Grid Graph</a>
%F A335586 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
%F A335586 a(n) ~ (1 + sqrt(2)) * exp(4*G*n^2/Pi), where G is Catalan's constant A006752. - _Vaclav Kotesovec_, Feb 14 2021
%e A335586 For n = 1, there are a(1) = 8 tilings (see the Links section for a diagram).
%o A335586 (PARI) default(realprecision, 120);
%o A335586 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)));
%o A335586 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)));
%o A335586 a(n) = if(n==0, 1, 4*b(n)+c(n)/2); \\ _Seiichi Manyama_, Feb 13 2021
%Y A335586 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).
%Y A335586 Cf. A004003, A212800, A340562, A341478, A341479.
%K A335586 nonn
%O A335586 0,2
%A A335586 _Drake Thomas_, Jan 26 2021
%E A335586 More terms from _Seiichi Manyama_, Feb 13 2021