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A368263 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to 180-degree rotation by an asymmetric tile.

%I A368263 #9 Oct 19 2024 21:54:51
%S A368263 1,3,2,4,7,2,10,20,16,4,16,76,88,43,4,36,272,720,538,120,9,64,1072,
%T A368263 5472,8356,3280,382,10,136,4160,43968,131464,105376,22028,1236,22,256,
%U A368263 16576,349568,2099728,3355456,1400536,149800,4243,30
%N A368263 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to 180-degree rotation by an asymmetric tile.
%H A368263 Peter Kagey, <a href="/A368263/a368263.pdf">Illustration of T(2,3)=20</a>
%H A368263 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 A368263 Table begins:
%e A368263   n\k| 1   2     3       4        5          6
%e A368263   ---+----------------------------------------
%e A368263    1 | 1   3     4      10       16         36
%e A368263    2 | 2   7    20      76      272       1072
%e A368263    3 | 2  16    88     720     5472      43968
%e A368263    4 | 4  43   538    8356   131464    2099728
%e A368263    5 | 4 120  3280  105376  3355456  107390592
%e A368263    6 | 9 382 22028 1400536 89489584 5726776672
%t A368263 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 A368263 nonn,tabl
%O A368263 1,2
%A A368263 _Peter Kagey_, Dec 21 2023