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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.
20

%I #39 Dec 04 2018 21:23:04

%S 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 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 2003,0,25,0,1,1,0,0,997,9157,0,0,9157,997,0,0,1

%N 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 Alois P. Heinz, <a href="/A230031/b230031.txt">Antidiagonals n = 0..20, flattened</a>

%H 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 R. S. Harris, <a href="http://www.bumblebeagle.org/polyominoes/tilingcounting">Counting Polyomino Tilings</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetris">Tetris</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetromino">Tetromino</a>

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

%e A(4,2) = A(2,4) = 4:

%e ._______. ._______. ._______. ._______.

%e | | | |_______| | |___. | | .___| |

%e |___|___| |_______| |_____|_| |_|_____|.

%e Square array A(n,k) begins:

%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

%e 1, 0, 0, 0, 1, 0, 0, 0, 1, ...

%e 1, 0, 1, 0, 4, 0, 9, 0, 25, ...

%e 1, 0, 0, 0, 23, 0, 0, 0, 997, ...

%e 1, 1, 4, 23, 117, 454, 2003, 9157, 40899, ...

%e 1, 0, 0, 0, 454, 0, 0, 0, 800290, ...

%e 1, 0, 9, 0, 2003, 0, 178939, 0, 22483347, ...

%e 1, 0, 0, 0, 9157, 0, 0, 0, 657253434, ...

%e 1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, ...

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

%Y Bisection of main diagonal (even part) gives A263425.

%Y Cf. A099390, A233320, A233427.

%K nonn,tabl

%O 0,24

%A _Alois P. Heinz_, Nov 29 2013