A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and trominoes (of any shape); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 6, 1, 1, 1, 2, 17, 30, 17, 2, 1, 1, 2, 43, 145, 145, 43, 2, 1, 1, 3, 108, 733, 1352, 733, 108, 3, 1, 1, 4, 280, 3540, 12688, 12688, 3540, 280, 4, 1, 1, 5, 727, 17300, 115958, 226922, 115958, 17300, 727, 5, 1

Offset

0,13

Links

Alois P. Heinz, Table of n, a(n) for n = 0..350

Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.

Wikipedia, Domino (mathematics)

Wikipedia, Tromino

Formula

A(n,k) = A(k,n).

Example

A(3,2) = A(2,3) = 6:
  .___.   .___.   .___.   .___.   .___.   .___.
  | | |   |___|   | | |   |___|   | ._|   |_. |
  | | |   |___|   |_|_|   | | |   |_| |   | |_|
  |_|_|   |___|   |___|   |_|_|   |___|   |___|  .
.
Square array A(n,k) begins:
  1, 1,   1,     1,       1,        1,          1,            1, ...
  1, 0,   1,     1,       1,        2,          2,            3, ...
  1, 1,   2,     6,      17,       43,        108,          280, ...
  1, 1,   6,    30,     145,      733,       3540,        17300, ...
  1, 1,  17,   145,    1352,    12688,     115958,      1075397, ...
  1, 2,  43,   733,   12688,   226922,    3927233,     68846551, ...
  1, 2, 108,  3540,  115958,  3927233,  128441094,   4263997124, ...
  1, 3, 280, 17300, 1075397, 68846551, 4263997124, 267855152858, ...

Crossrefs

Columns (or rows) k=0-10 give: A000012, A182097(n) = A000931(n+3), A019439, A364460, A364155, A364556, A364616, A364617, A364632, A364638, A364640.

Main diagonal gives A364504.

Cf. A219866, A219987, A233320.

Sequence in context: A247005 A380164 A174215 * A305567 A134744 A394909

Adjacent sequences: A364454 A364455 A364456 * A364458 A364459 A364460

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 25 2023

Extensions

Terms n,k>=4 had to be corrected as was pointed out by Martin Fuller and David Radcliffe - Alois P. Heinz, Apr 05 2025

Status

approved