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%I #8 Dec 21 2023 15:14:13
%S 1,2,2,4,6,2,8,20,12,4,16,72,88,39,4,32,272,688,538,104,9,64,1056,
%T 5472,8292,3280,366,10,128,4160,43712,131464,104864,22028,1172,22,256,
%U 16512,349568,2098704,3355456,1399512,149800,4179,30
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k cylinder up to horizontal reflection by an asymmetric tile.
%H Peter Kagey, <a href="/A368259/a368259.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 2 4 8 16 32
%e 2 | 2 6 20 72 272 1056
%e 3 | 2 12 88 688 5472 43712
%e 4 | 4 39 538 8292 131464 2098704
%e 5 | 4 104 3280 104864 3355456 107374208
%e 6 | 9 366 22028 1399512 89489584 5726711136
%t A368259[n_,m_]:=1/(2n) (DivisorSum[n,EulerPhi[#]*2^(n*m/#)&]+n*2^(n*m/2-1)*Boole[EvenQ[n]])
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023
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