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A368262 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 two tiles that are each fixed under 180-degree rotation.

%I A368262 #9 Oct 19 2024 21:54:51
%S A368262 2,3,3,6,7,4,10,24,16,6,20,76,104,43,8,36,288,720,570,120,13,72,1072,
%T A368262 5600,8356,3408,382,18,136,4224,43968,131976,105376,22284,1236,30,272,
%U A368262 16576,350592,2099728,3359552,1400536,150824,4243,46
%N A368262 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 two tiles that are each fixed under 180-degree rotation.
%H A368262 Peter Kagey, <a href="/A368262/a368262.pdf">Illustration of T(2,3)=24</a>
%H A368262 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 A368262 Table begins:
%e A368262   n\k|  1   2     3       4        5          6
%e A368262   ---+-----------------------------------------
%e A368262    1 |  2   3     6      10       20         36
%e A368262    2 |  3   7    24      76      288       1072
%e A368262    3 |  4  16   104     720     5600      43968
%e A368262    4 |  6  43   570    8356   131976    2099728
%e A368262    5 |  8 120  3408  105376  3359552  107390592
%e A368262    6 | 13 382 22284 1400536 89505968 5726776672
%t A368262 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 A368262 nonn,tabl
%O A368262 1,1
%A A368262 _Peter Kagey_, Dec 21 2023