A341741 - OEIS

%I #52 Dec 27 2021 11:00:19

%S 2,8,2,14,36,2,36,50,200,2,82,272,224,1156,2,200,722,3108,1058,6728,2,

%T 478,3108,9922,39952,5054,39204,2,1156,10082,90176,155682,537636,

%U 24200,228488,2,2786,39952,401998,3113860,2540032,7379216,115934,1331716,2

%N Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

%C Dimer tilings of 2n x k toroidal grid.

%H Seiichi Manyama, <a href="/A341741/b341741.txt">Antidiagonals n = 1..50, flattened</a>

%H P. W. Kasteleyn, <a href="https://doi.org/10.1016/0031-8914(61)90063-5">The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice</a>, Physica 27 (1961), 1209-1225. See Eq. (25).

%H <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>

%F T(n,k) = A341533(n,k)/2 + A341738(n,k) + 2 * ((k+1) mod 2) * A341739(n,ceiling(k/2)).

%F T(n, 2k) = T(k, 2n).

%F If k is odd, T(n,k) = A341533(n,k) = 2*A341738(n,k).

%e Square array begins:

%e   2,     8,    14,      36,       82,        200, ...

%e   2,    36,    50,     272,      722,       3108, ...

%e   2,   200,   224,    3108,     9922,      90176, ...

%e   2,  1156,  1058,   39952,   155682,    3113860, ...

%e   2,  6728,  5054,  537636,  2540032,  114557000, ...

%e   2, 39204, 24200, 7379216, 41934482, 4357599552, ...

%Y Columns 1..12 give A007395, A162484(2*n), A231087, A220864(2*n), A231485, A232804(2*n), A230033, A253678(2*n), A281583, A281679(2*n), A308761, A309018(2*n).

%Y Rows 1..6 give A162484, A220864, A232804, A253678, A281679, A309018.

%Y T(n,2*n) gives A335586.

%Y Cf. A341533, A341738, A341739.

%Y Cf. A099390, A103997, A103999.

%K nonn,tabl

%O 1,1

%A _Seiichi Manyama_, Feb 18 2021

%E New name from _Andrey Zabolotskiy_, Dec 26 2021