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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 an asymmetric tile.
5

%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