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A368221 Table read by downward antidiagonals: 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.

%I A368221 #15 Oct 19 2024 21:54:51
%S A368221 2,4,3,8,10,6,16,36,40,10,32,136,288,136,20,64,528,2176,2080,544,36,
%T A368221 128,2080,16896,32896,16640,2080,72,256,8256,133120,524800,526336,
%U A368221 131328,8320,136,512,32896,1056768,8390656,16793600,8390656,1050624,32896,272
%N A368221 Table read by downward antidiagonals: 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 A368221 Peter Kagey, <a href="/A368221/a368221_1.pdf">Illustration of T(3,2)=40</a>
%H A368221 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 A368221 Table begins:
%e A368221   n\k|  1    2      3       4         5           6
%e A368221   ---+---------------------------------------------
%e A368221    1 |  2    4      8      16        32          64
%e A368221    2 |  3   10     36     136       528        2080
%e A368221    3 |  6   40    288    2176     16896      133120
%e A368221    4 | 10  136   2080   32896    524800     8390656
%e A368221    5 | 20  544  16640  526336  16793600   537001984
%e A368221    6 | 36 2080 131328 8390656 536887296 34359869440
%t A368221 A368221[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n], 2^(n*m/2), 2^(m (n + 1)/2)])
%Y A368221 Cf. A368218, A368222.
%K A368221 nonn,tabl
%O A368221 1,1
%A A368221 _Peter Kagey_, Dec 18 2023