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%I #8 Dec 21 2023 15:14:33
%S 1,3,2,4,7,2,10,20,14,4,16,76,88,40,4,36,272,700,532,108,8,64,1072,
%T 5472,8296,3280,362,10,136,4160,43800,131344,104968,21944,1182,20,256,
%U 16576,349568,2098720,3355456,1399176,149800,4150,30
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflection by an asymmetric tile.
%H Peter Kagey, <a href="/A368261/a368261.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 14 88 700 5472 43800
%e 4 | 4 40 532 8296 131344 2098720
%e 5 | 4 108 3280 104968 3355456 107377488
%e 6 | 8 362 21944 1399176 89484128 5726689312
%t A368261[n_, m_]:=1/(2n)*(DivisorSum[n, EulerPhi[#]*2^(n*m/#)&] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2])&], DivisorSum[n, EulerPhi[#]*2^(n*m/#)&, EvenQ]])
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023
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