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%I #15 Dec 21 2023 11:03:10
%S 1,2,2,3,7,3,6,20,20,6,10,76,136,76,10,20,272,1056,1056,272,20,36,
%T 1072,8256,16576,8256,1072,36,72,4160,65792,262656,262656,65792,4160,
%U 72,136,16576,524800,4197376,8390656,4197376,524800,16576,136
%N Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.
%H Peter Kagey, <a href="/A368219/a368219.pdf">Illustration of T(3,2)=20</a>
%H Peter Kagey and William Keehn, <a href="https://arxiv.org/abs/2311.13072">Counting tilings of the n X m grid, cylinder, and torus</a>, arXiv: 2311.13072 [math.CO], 2023.
%e Table begins:
%e n\k | 1 2 3 4 5 6
%e ----+--------------------------------------------
%e 1 | 1 2 3 6 10 20
%e 2 | 2 7 20 76 272 1072
%e 3 | 3 20 136 1056 8256 65792
%e 4 | 6 76 1056 16576 262656 4197376
%e 5 | 10 272 8256 262656 8390656 268451840
%e 6 | 20 1072 65792 4197376 268451840 17180065792
%t A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])
%Y Cf. A225910, A368218, A368220, A368223.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 18 2023
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