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%I #11 Dec 21 2023 21:12:33
%S 1,2,2,2,5,2,4,8,8,4,4,22,24,22,4,8,44,120,120,44,8,9,135,612,1203,
%T 612,135,9,18,362,3892,13600,13600,3892,362,18,23,1211,25482,177342,
%U 337600,177342,25482,1211,23,44,3914,176654,2404372,8962618,8962618,2404372,176654,3914,44
%N Table read by antidiagonals downward: T(n,k) is the number of tilings of the n X k torus up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotations but not horizontal or vertical reflections.
%H Peter Kagey, <a href="/A368303/a368303.pdf">Illustration of T(3,3)=24</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 2 4 4 8
%e 2 | 2 5 8 22 44 135
%e 3 | 2 8 24 120 612 3892
%e 4 | 4 22 120 1203 13600 177342
%e 5 | 4 44 612 13600 337600 8962618
%e 6 | 8 135 3892 177342 8962618 477371760
%t A368303[n_, m_]:=1/(4*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, Function[c, EulerPhi[c]EulerPhi[d]2^(m*n/LCM[c, d])]]]] + If[EvenQ[n], n/2*DivisorSum[m, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*2^(2m/#)*Boole[EvenQ[#]])&], n*DivisorSum[m, EulerPhi[#](2^(n*m/#))&, EvenQ]] + If[EvenQ[m], m/2*DivisorSum[n, EulerPhi[#](2^(n*m/LCM[2, #]) + 2^((m - 2)*n/LCM[2, #])*2^(2n/#)*Boole[EvenQ[#]])&], m*DivisorSum[n, EulerPhi[#](2^(m*n/#))&, EvenQ]] + n*m*2^((n*m)/2)*Which[OddQ[n*m], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])
%Y Cf. A222188, A368219, A368256, A368302, A368304, A368307.
%K nonn,tabl
%O 1,2
%A _Peter Kagey_, Dec 21 2023
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