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Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.
4

%I #15 Dec 21 2023 11:04:39

%S 2,3,3,6,10,6,10,36,36,10,20,136,272,136,20,36,528,2080,2080,528,36,

%T 72,2080,16512,32896,16512,2080,72,136,8256,131328,524800,524800,

%U 131328,8256,136,272,32896,1049600,8390656,16781312,8390656,1049600,32896,272

%N Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by two tiles that are each fixed under 180-degree rotation.

%H Peter Kagey, <a href="/A368223/a368223.pdf">Illustration of T(3,3)=272</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 | 2 3 6 10 20 36

%e 2 | 3 10 36 136 528 2080

%e 3 | 6 36 272 2080 16512 131328

%e 4 | 10 136 2080 32896 524800 8390656

%e 5 | 20 528 16512 524800 16781312 536887296

%e 6 | 36 2080 131328 8390656 536887296 34359869440

%t A368223[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n*m], 2^(n*m/2), 2^((n*m + 1)/2)])

%Y Cf. A368219, A368224.

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 18 2023