%I #13 Dec 21 2023 11:03:55
%S 1,3,3,4,10,4,10,36,36,10,16,136,256,136,16,36,528,2080,2080,528,36,
%T 64,2080,16384,32896,16384,2080,64,136,8256,131328,524800,524800,
%U 131328,8256,136,256,32896,1048576,8390656,16777216,8390656,1048576,32896,256
%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 an asymmetric tile.
%H Peter Kagey, <a href="/A368224/a368224.pdf">Illustration of T(3,3)=256</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 3 4 10 16 36
%e 2 | 3 10 36 136 528 2080
%e 3 | 4 36 256 2080 16384 131328
%e 4 | 10 136 2080 32896 524800 8390656
%e 5 | 16 528 16384 524800 16777216 536887296
%e 6 | 36 2080 131328 8390656 536887296 34359869440
%t A368224[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n*m]])
%Y Cf. A368220, A368222, A368223, A117401.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 18 2023