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A368224 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.

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