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%I #14 Dec 21 2023 11:05:14
%S 2,4,3,8,10,6,16,36,40,10,32,136,288,136,20,64,528,2176,2080,544,36,
%T 128,2080,16896,32896,16640,2080,72,256,8256,133120,524800,526336,
%U 131328,8320,136,512,32896,1056768,8390656,16793600,8390656,1050624,32896,272
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by two tiles that are each fixed under horizontal reflection.
%H Peter Kagey, <a href="/A368221/a368221_1.pdf">Illustration of T(3,2)=40</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 4 8 16 32 64
%e 2 | 3 10 36 136 528 2080
%e 3 | 6 40 288 2176 16896 133120
%e 4 | 10 136 2080 32896 524800 8390656
%e 5 | 20 544 16640 526336 16793600 537001984
%e 6 | 36 2080 131328 8390656 536887296 34359869440
%t A368221[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n], 2^(n*m/2), 2^(m (n + 1)/2)])
%Y Cf. A368218, A368222.
%K nonn,tabl
%O 1,1
%A _Peter Kagey_, Dec 18 2023
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