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A368261 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflection by an asymmetric tile.

%I A368261 #9 Oct 19 2024 21:54:51
%S A368261 1,3,2,4,7,2,10,20,14,4,16,76,88,40,4,36,272,700,532,108,8,64,1072,
%T A368261 5472,8296,3280,362,10,136,4160,43800,131344,104968,21944,1182,20,256,
%U A368261 16576,349568,2098720,3355456,1399176,149800,4150,30
%N A368261 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to vertical reflection by an asymmetric tile.
%H A368261 Peter Kagey, <a href="/A368261/a368261.pdf">Illustration of T(2,3)=20</a>
%H A368261 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 A368261 Table begins:
%e A368261   n\k| 1   2     3       4        5          6
%e A368261   ---+----------------------------------------
%e A368261    1 | 1   3     4      10       16         36
%e A368261    2 | 2   7    20      76      272       1072
%e A368261    3 | 2  14    88     700     5472      43800
%e A368261    4 | 4  40   532    8296   131344    2098720
%e A368261    5 | 4 108  3280  104968  3355456  107377488
%e A368261    6 | 8 362 21944 1399176 89484128 5726689312
%t A368261 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 A368261 nonn,tabl
%O A368261 1,2
%A A368261 _Peter Kagey_, Dec 21 2023