A341741 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.

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 272, 224, 1156, 2, 200, 722, 3108, 1058, 6728, 2, 478, 3108, 9922, 39952, 5054, 39204, 2, 1156, 10082, 90176, 155682, 537636, 24200, 228488, 2, 2786, 39952, 401998, 3113860, 2540032, 7379216, 115934, 1331716, 2

Offset

1,1

Comments

Dimer tilings of 2n x k toroidal grid.

Links

Seiichi Manyama, Antidiagonals n = 1..50, flattened

P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. See Eq. (25).

Index entries for sequences related to dominoes

Formula

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

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

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

Example

Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     272,      722,       3108, ...
  2,   200,   224,    3108,     9922,      90176, ...
  2,  1156,  1058,   39952,   155682,    3113860, ...
  2,  6728,  5054,  537636,  2540032,  114557000, ...
  2, 39204, 24200, 7379216, 41934482, 4357599552, ...

Crossrefs

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).

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

T(n,2*n) gives A335586.

Cf. A341533, A341738, A341739.

Cf. A099390, A103997, A103999.

Sequence in context: A222842 A098471 A341533 * A074723 A286455 A175183

Adjacent sequences: A341738 A341739 A341740 * A341742 A341743 A341744

Keyword

nonn,tabl

Author

Seiichi Manyama, Feb 18 2021

Extensions

New name from Andrey Zabolotskiy, Dec 26 2021

Status

approved