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Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.
2

%I #10 Oct 19 2024 21:54:51

%S 2,4,3,8,10,4,16,36,20,6,32,136,120,55,8,64,528,816,666,136,13,128,

%T 2080,5984,9316,3536,430,18,256,8256,45760,139656,106912,23052,1300,

%U 30,512,32896,357760,2164240,3371840,1415896,151848,4435,46

%N Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflections by two tiles that are each fixed under horizontal reflection.

%H Peter Kagey, <a href="/A368258/a368258.pdf">Illustration of T(2,3)=36</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 | 4 20 120 816 5984 45760

%e 4 | 6 55 666 9316 139656 2164240

%e 5 | 8 136 3536 106912 3371840 107505280

%e 6 | 13 430 23052 1415896 89751728 5730905440

%e 7 | 18 1300 151848 19206736 2454791328 314154568000

%t A368258[n_,m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + n*2^(n*m/2)*If[EvenQ[n], 1/2*(2^m + 1), 2^(m/2)])

%K nonn,tabl

%O 1,1

%A _Peter Kagey_, Dec 21 2023