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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.

%I A230031 #48 Mar 26 2025 16:23:18
%S A230031 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,
%T A230031 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,
%U A230031 2003,0,25,0,1,1,0,0,997,9157,0,0,9157,997,0,0,1
%N 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.
%H A230031 Liang Kai, <a href="/A230031/b230031.txt">Antidiagonals n = 0..27, flattened</a> (Antidiagonals n = 0..20 from Alois P. Heinz)
%H A230031 S. Butler, J. Ekstrand, S. Osborne, <a href="/A230031/a230031.pdf">TETRIS Tiling</a>, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
%H A230031 R. S. Harris, <a href="http://www.bumblebeagle.org/polyominoes/tilingcounting">Counting Polyomino Tilings</a>
%H A230031 Liang Kai, <a href="https://arxiv.org/abs/2503.17698">Solving tiling enumeration problems by tensor network contractions</a>, arXiv:2503.17698 [math.CO], 2025.
%H A230031 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetris">Tetris</a>
%H A230031 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>
%F A230031 A(n,k) = 0 <=> n*k mod 4 > 0.
%e A230031 A(4,2) = A(2,4) = 4:
%e A230031   ._______.  ._______.  ._______.  ._______.
%e A230031   |   |   |  |_______|  | |___. |  | .___| |
%e A230031   |___|___|  |_______|  |_____|_|  |_|_____|.
%e A230031 Square array A(n,k) begins:
%e A230031   1, 1,  1,   1,     1,      1,        1,         1,           1, ...
%e A230031   1, 0,  0,   0,     1,      0,        0,         0,           1, ...
%e A230031   1, 0,  1,   0,     4,      0,        9,         0,          25, ...
%e A230031   1, 0,  0,   0,    23,      0,        0,         0,         997, ...
%e A230031   1, 1,  4,  23,   117,    454,     2003,      9157,       40899, ...
%e A230031   1, 0,  0,   0,   454,      0,        0,         0,      800290, ...
%e A230031   1, 0,  9,   0,  2003,      0,   178939,         0,    22483347, ...
%e A230031   1, 0,  0,   0,  9157,      0,        0,         0,   657253434, ...
%e A230031   1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, ...
%Y A230031 Columns (or rows) include: A000012, A007598, A232757, A174248, A232758, A232684, A232759, A232698, A247113, A232722.
%Y A230031 Bisection of main diagonal (even part) gives A263425.
%Y A230031 Cf. A099390, A233320, A233427.
%K A230031 nonn,tabl
%O A230031 0,24
%A A230031 _Alois P. Heinz_, Nov 29 2013