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%I #8 Dec 21 2023 15:14:40
%S 2,3,3,6,7,4,10,24,16,6,20,76,104,43,8,36,288,720,570,120,13,72,1072,
%T 5600,8356,3408,382,18,136,4224,43968,131976,105376,22284,1236,30,272,
%U 16576,350592,2099728,3359552,1400536,150824,4243,46
%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 two tiles that are each fixed under 180-degree rotation.
%H Peter Kagey, <a href="/A368262/a368262.pdf">Illustration of T(2,3)=24</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 3 6 10 20 36
%e 2 | 3 7 24 76 288 1072
%e 3 | 4 16 104 720 5600 43968
%e 4 | 6 43 570 8356 131976 2099728
%e 5 | 8 120 3408 105376 3359552 107390592
%e 6 | 13 382 22284 1400536 89505968 5726776672
%t A368262[n_, m_] := 1/(2n)*(DivisorSum[n, Function[d, EulerPhi[d]*2^(n*m/d)]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])
%K nonn,tabl
%O 1,1
%A _Peter Kagey_, Dec 21 2023
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