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A368222 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

%I A368222 #13 Oct 19 2024 21:54:51
%S A368222 1,2,3,4,10,4,8,36,32,10,16,136,256,136,16,32,528,2048,2080,512,36,64,
%T A368222 2080,16384,32896,16384,2080,64,128,8256,131072,524800,524288,131328,
%U A368222 8192,136,256,32896,1048576,8390656,16777216,8390656,1048576,32896,256
%N A368222 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.
%H A368222 Peter Kagey, <a href="/A368222/a368222.pdf">Illustration of T(3,2)=32</a>
%H A368222 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 A368222 Table begins:
%e A368222   n\k|  1    2      3       4         5           6
%e A368222   ---+---------------------------------------------
%e A368222    1 |  1    2      4       8        16          32
%e A368222    2 |  3   10     36     136       528        2080
%e A368222    3 |  4   32    256    2048     16384      131072
%e A368222    4 | 10  136   2080   32896    524800     8390656
%e A368222    5 | 16  512  16384  524288  16777216   536870912
%e A368222    6 | 36 2080 131328 8390656 536887296 34359869440
%t A368222 A368222[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n]])
%Y A368222 Cf. A368220, A368221, A368224, A117401.
%K A368222 nonn,tabl
%O A368222 1,2
%A A368222 _Peter Kagey_, Dec 18 2023