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%I #8 Dec 21 2023 15:14:50
%S 1,3,2,4,7,2,10,20,16,4,16,76,88,43,4,36,272,720,538,120,9,64,1072,
%T 5472,8356,3280,382,10,136,4160,43968,131464,105376,22028,1236,22,256,
%U 16576,349568,2099728,3355456,1400536,149800,4243,30
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to 180-degree rotation by an asymmetric tile.
%H Peter Kagey, <a href="/A368263/a368263.pdf">Illustration of T(2,3)=20</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 | 1 3 4 10 16 36
%e 2 | 2 7 20 76 272 1072
%e 3 | 2 16 88 720 5472 43968
%e 4 | 4 43 538 8356 131464 2099728
%e 5 | 4 120 3280 105376 3355456 107390592
%e 6 | 9 382 22028 1400536 89489584 5726776672
%t A368263[n_, m_] := 1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 1/2, True, 0])
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023
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