210223

PARI


Number of perfect matchings in the graph C_{2n} X C_k

kが3以上のとき、計算してみた。

a341533(n, k) = round(sqrt(prod(a=1, n,   prod(b=1, k,   4*sin((2*a-1)*Pi/(2*n))^2+4*sin((2*b-1)*Pi/k)^2))));
a341738(n, k) = round(sqrt(prod(a=1, n,   prod(b=1, k-1, 4*sin((2*a-1)*Pi/(2*n))^2+4*sin(2*b*Pi/k)^2))));
a341739(n, k) = round(     prod(a=1, n-1, prod(b=1, k,   4*sin(a*Pi/n)^2          +4*sin((2*b-1)*Pi/(2*k))^2)));
T(n, k) = a341533(n, k)/2+a341738(n, k)+2*((k+1)%2)*a341739(n, ceil(k/2));

K=7;
for(n=2, 10, for(k=3, K, print1(T(n, k), ", ")); print)

出力結果

50, 272, 722, 3108, 10082, 

224, 3108, 9922, 90176, 401998, 

1058, 39952, 155682, 3113860, 19681538, 

5054, 537636, 2540032, 114557000, 1034315998, 

24200, 7379216, 41934482, 4357599552, 55820091938, 

115934, 102144036, 694861522, 169094614280, 3044533460992, 

555458, 1418981392, 11527389122, 6640849944580, 166779871224962, 

2661344, 19742287908, 191304901282, 262671237617216, 9152970837103102, 

12751250, 274848860432, 3175220160032, 10433179552323108, 502711247500143362,