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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.
4

%I #15 Oct 19 2024 21:54:51

%S 2,4,3,8,10,6,16,36,40,10,32,136,288,136,20,64,528,2176,2080,544,36,

%T 128,2080,16896,32896,16640,2080,72,256,8256,133120,524800,526336,

%U 131328,8320,136,512,32896,1056768,8390656,16793600,8390656,1050624,32896,272

%N 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 Peter Kagey, <a href="/A368221/a368221_1.pdf">Illustration of T(3,2)=40</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 | 2 4 8 16 32 64

%e 2 | 3 10 36 136 528 2080

%e 3 | 6 40 288 2176 16896 133120

%e 4 | 10 136 2080 32896 524800 8390656

%e 5 | 20 544 16640 526336 16793600 537001984

%e 6 | 36 2080 131328 8390656 536887296 34359869440

%t A368221[n_, m_] := 1/2 (2^(n*m) + If[EvenQ[n], 2^(n*m/2), 2^(m (n + 1)/2)])

%Y Cf. A368218, A368222.

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 18 2023