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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.
3

%I #10 Oct 19 2024 21:54:51

%S 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 4244,1704,205,9,72,2144,22028,66184,52740,11228,623,18,136,8384,

%U 175296,1050896,1679776,701124,75412,2171,23

%N 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 Peter Kagey, <a href="/A368256/a368256.pdf">Illustration of T(2,3)=14</a>

%H 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 Table begins:

%e n\k| 1 2 3 4 5 6

%e ---+--------------------------------------------

%e 1 | 1 2 3 6 10 20

%e 2 | 2 5 14 44 152 560

%e 3 | 2 9 52 366 2800 22028

%e 4 | 4 26 298 4244 66184 1050896

%e 5 | 4 62 1704 52740 1679776 53696936

%e 6 | 8 205 11228 701124 44758448 2863442960

%e 7 | 9 623 75412 9591666 1227199056 157073688884

%t 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 nonn,tabl

%O 1,2

%A _Peter Kagey_, Dec 21 2023