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A368260 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflections by two tiles that are each fixed under vertical reflection.

%I A368260 #9 Oct 19 2024 21:54:51
%S A368260 2,3,3,6,7,4,10,24,14,6,20,76,100,40,8,36,288,700,564,108,14,72,1072,
%T A368260 5560,8296,3384,362,20,136,4224,43800,131856,104968,22288,1182,36,272,
%U A368260 16576,350256,2098720,3358736,1399176,150972,4150,60
%N A368260 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflections by two tiles that are each fixed under vertical reflection.
%H A368260 Peter Kagey, <a href="/A368260/a368260.pdf">Illustration of T(2,3)=24</a>
%H A368260 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 A368260 Table begins:
%e A368260   n\k|  1   2     3       4        5          6
%e A368260   ---+-----------------------------------------
%e A368260    1 |  2   3     6      10       20         36
%e A368260    2 |  3   7    24      76      288       1072
%e A368260    3 |  4  14   100     700     5560      43800
%e A368260    4 |  6  40   564    8296   131856    2098720
%e A368260    5 |  8 108  3384  104968  3358736  107377488
%e A368260    6 | 14 362 22288 1399176 89505984 5726689312
%t A368260 A368260[n_, m_] := 1/(2 n) (DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + If[EvenQ[m], DivisorSum[n,  EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n,  EulerPhi[#]*2^((n*m - n)/LCM[#, 2])*2^(n/#) &]])
%K A368260 nonn,tabl
%O A368260 1,1
%A A368260 _Peter Kagey_, Dec 21 2023