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A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and trominoes (of any shape); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A364457 #65 Apr 05 2025 15:43:55
%S A364457 1,1,1,1,0,1,1,1,1,1,1,1,2,1,1,1,1,6,6,1,1,1,2,17,30,17,2,1,1,2,43,
%T A364457 145,145,43,2,1,1,3,108,733,1352,733,108,3,1,1,4,280,3540,12688,12688,
%U A364457 3540,280,4,1,1,5,727,17300,115958,226922,115958,17300,727,5,1
%N A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and trominoes (of any shape); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A364457 Alois P. Heinz, <a href="/A364457/b364457.txt">Table of n, a(n) for n = 0..350</a>
%H A364457 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 A364457 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_(mathematics)">Domino (mathematics)</a>
%H A364457 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tromino">Tromino</a>
%F A364457 A(n,k) = A(k,n).
%e A364457 A(3,2) = A(2,3) = 6:
%e A364457   .___.   .___.   .___.   .___.   .___.   .___.
%e A364457   | | |   |___|   | | |   |___|   | ._|   |_. |
%e A364457   | | |   |___|   |_|_|   | | |   |_| |   | |_|
%e A364457   |_|_|   |___|   |___|   |_|_|   |___|   |___|  .
%e A364457 .
%e A364457 Square array A(n,k) begins:
%e A364457   1, 1,   1,     1,       1,        1,          1,            1, ...
%e A364457   1, 0,   1,     1,       1,        2,          2,            3, ...
%e A364457   1, 1,   2,     6,      17,       43,        108,          280, ...
%e A364457   1, 1,   6,    30,     145,      733,       3540,        17300, ...
%e A364457   1, 1,  17,   145,    1352,    12688,     115958,      1075397, ...
%e A364457   1, 2,  43,   733,   12688,   226922,    3927233,     68846551, ...
%e A364457   1, 2, 108,  3540,  115958,  3927233,  128441094,   4263997124, ...
%e A364457   1, 3, 280, 17300, 1075397, 68846551, 4263997124, 267855152858, ...
%Y A364457 Columns (or rows) k=0-10 give: A000012, A182097(n) = A000931(n+3), A019439, A364460, A364155, A364556, A364616, A364617, A364632, A364638, A364640.
%Y A364457 Main diagonal gives A364504.
%Y A364457 Cf. A219866, A219987, A233320.
%K A364457 nonn,tabl
%O A364457 0,13
%A A364457 _Alois P. Heinz_, Jul 25 2023
%E A364457 Terms n,k>=4 had to be corrected as was pointed out by _Martin Fuller_ and _David Radcliffe_ - _Alois P. Heinz_, Apr 05 2025