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A368256 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflections.

%I A368256 #10 Oct 19 2024 21:54:51
%S A368256 1,2,2,3,5,2,6,14,9,4,10,44,52,26,4,20,152,366,298,62,8,36,560,2800,
%T A368256 4244,1704,205,9,72,2144,22028,66184,52740,11228,623,18,136,8384,
%U A368256 175296,1050896,1679776,701124,75412,2171,23
%N A368256 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflections.
%H A368256 Peter Kagey, <a href="/A368256/a368256.pdf">Illustration of T(2,3)=14</a>
%H A368256 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 A368256 Table begins:
%e A368256   n\k| 1   2     3       4          5            6
%e A368256   ---+--------------------------------------------
%e A368256    1 | 1   2     3       6         10           20
%e A368256    2 | 2   5    14      44        152          560
%e A368256    3 | 2   9    52     366       2800        22028
%e A368256    4 | 4  26   298    4244      66184      1050896
%e A368256    5 | 4  62  1704   52740    1679776     53696936
%e A368256    6 | 8 205 11228  701124   44758448   2863442960
%e A368256    7 | 9 623 75412 9591666 1227199056 157073688884
%t A368256 A368256[n_, m_] := 1/(4n)*( DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n (2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*2^(n*m/#) &, EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])
%K A368256 nonn,tabl
%O A368256 1,2
%A A368256 _Peter Kagey_, Dec 21 2023