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

Author

Alois P. Heinz, Nov 29 2013

Keywords

• nonn
• tabl

Examples

			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, ...
		

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

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.

Formula

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