A230031 Number A(n,k) of tilings of a k X n rectangle using tetrominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 4, 0, 4, 0, 1, 1, 0, 0, 23, 23, 0, 0, 1, 1, 0, 9, 0, 117, 0, 9, 0, 1, 1, 1, 0, 0, 454, 454, 0, 0, 1, 1, 1, 0, 25, 0, 2003, 0, 2003, 0, 25, 0, 1, 1, 0, 0, 997, 9157, 0, 0, 9157, 997, 0, 0, 1

Offset

0,24

Links

Liang Kai, Antidiagonals n = 0..27, flattened (Antidiagonals n = 0..20 from Alois P. Heinz)

S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013

R. S. Harris, Counting Polyomino Tilings

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

Wikipedia, Tetris

Wikipedia, Tetromino

Formula

A(n,k) = 0 <=> n*k mod 4 > 0.

Example

A(4,2) = A(2,4) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |_______|  | |___. |  | .___| |
  |___|___|  |_______|  |_____|_|  |_|_____|.
Square array A(n,k) begins:
  1, 1,  1,   1,     1,      1,        1,         1,           1, ...
  1, 0,  0,   0,     1,      0,        0,         0,           1, ...
  1, 0,  1,   0,     4,      0,        9,         0,          25, ...
  1, 0,  0,   0,    23,      0,        0,         0,         997, ...
  1, 1,  4,  23,   117,    454,     2003,      9157,       40899, ...
  1, 0,  0,   0,   454,      0,        0,         0,      800290, ...
  1, 0,  9,   0,  2003,      0,   178939,         0,    22483347, ...
  1, 0,  0,   0,  9157,      0,        0,         0,   657253434, ...
  1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, ...

Crossrefs

Columns (or rows) include: A000012, A007598, A232757, A174248, A232758, A232684, A232759, A232698, A247113, A232722.

Bisection of main diagonal (even part) gives A263425.

Cf. A099390, A233320, A233427.

Sequence in context: A147986 A147988 A306488 * A391753 A019920 A246130

Adjacent sequences: A230028 A230029 A230030 * A230032 A230033 A230034

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 29 2013

Status

approved