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A368223 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.

%I A368223 #15 Dec 21 2023 11:04:39
%S A368223 2,3,3,6,10,6,10,36,36,10,20,136,272,136,20,36,528,2080,2080,528,36,
%T A368223 72,2080,16512,32896,16512,2080,72,136,8256,131328,524800,524800,
%U A368223 131328,8256,136,272,32896,1049600,8390656,16781312,8390656,1049600,32896,272
%N A368223 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 A368223 Peter Kagey, <a href="/A368223/a368223.pdf">Illustration of T(3,3)=272</a>
%H A368223 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 A368223 Table begins:
%e A368223   n\k|  1    2      3       4         5           6
%e A368223   ---+---------------------------------------------
%e A368223    1 |  2    3      6      10        20          36
%e A368223    2 |  3   10     36     136       528        2080
%e A368223    3 |  6   36    272    2080     16512      131328
%e A368223    4 | 10  136   2080   32896    524800     8390656
%e A368223    5 | 20  528  16512  524800  16781312   536887296
%e A368223    6 | 36 2080 131328 8390656 536887296 34359869440
%t A368223 A368223[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n*m], 2^(n*m/2), 2^((n*m + 1)/2)])
%Y A368223 Cf. A368219, A368224.
%K A368223 nonn,tabl
%O A368223 1,1
%A A368223 _Peter Kagey_, Dec 18 2023