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%I #9 Dec 21 2023 15:14:01
%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 antidiagonals downward: 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
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