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A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A233427 #52 Mar 26 2025 16:26:50
%S A233427 1,1,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,5,0,0,5,
%T A233427 0,1,1,0,0,56,0,56,0,0,1,1,0,0,0,501,501,0,0,0,1,1,0,0,0,0,4006,0,0,0,
%U A233427 0,1,1,1,0,0,0,27950,27950,0,0,0,1,1,1,0,45,0,0,214689,0,214689,0,0,45,0,1
%N A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A233427 Liang Kai, <a href="/A233427/b233427.txt">Antidiagonals n = 0..26, flattened</a> (Antidiagonals n = 0..17 from Alois P. Heinz)
%H A233427 R. S. Harris, <a href="http://www.bumblebeagle.org/polyominoes/tilingcounting">Counting Polyomino Tilings</a>
%H A233427 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 A233427 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pentomino">Pentomino</a>
%F A233427 A(n,k) = 0 <=> n*k mod 5 > 0.
%e A233427 A(5,2) = A(2,5) = 5:
%e A233427   ._________. ._________. ._________. ._________. ._________.
%e A233427   |_________| | ._____| | | |_____. | |   ._|   | |   |_.   |
%e A233427   |_________| |_|_______| |_______|_| |___|_____| |_____|___|.
%e A233427 Square array A(n,k) begins:
%e A233427   1, 1,  1,    1,      1,         1,          1, ...
%e A233427   1, 0,  0,    0,      0,         1,          0, ...
%e A233427   1, 0,  0,    0,      0,         5,          0, ...
%e A233427   1, 0,  0,    0,      0,        56,          0, ...
%e A233427   1, 0,  0,    0,      0,       501,          0, ...
%e A233427   1, 1,  5,   56,    501,      4006,      27950, ...
%e A233427   1, 0,  0,    0,      0,     27950,          0, ...
%e A233427   1, 0,  0,    0,      0,    214689,          0, ...
%e A233427   1, 0,  0,    0,      0,   1696781,          0, ...
%e A233427   1, 0,  0,    0,      0,  13205354,          0, ...
%e A233427   1, 1, 45, 7670, 890989, 101698212, 7845888732, ...
%e A233427   ...
%Y A233427 Columns (or rows) include: A000012, A054318, A233428, A233429, A174249, A233430.
%Y A233427 Cf. A099390, A233320, A230031, A246902, A247117, A278657.
%Y A233427 Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713 give A(n,5).
%K A233427 nonn,tabl
%O A233427 0,31
%A A233427 _Alois P. Heinz_, Dec 09 2013