A368303 Table read by downward antidiagonals: 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.
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, 612, 135, 9, 18, 362, 3892, 13600, 13600, 3892, 362, 18, 23, 1211, 25482, 177342, 337600, 177342, 25482, 1211, 23, 44, 3914, 176654, 2404372, 8962618, 8962618, 2404372, 176654, 3914, 44
Offset: 1
Examples
Table begins: n\k| 1 2 3 4 5 6 ---+------------------------------------ 1 | 1 2 2 4 4 8 2 | 2 5 8 22 44 135 3 | 2 8 24 120 612 3892 4 | 4 22 120 1203 13600 177342 5 | 4 44 612 13600 337600 8962618 6 | 8 135 3892 177342 8962618 477371760
Links
- Peter Kagey, Illustration of T(3,3)=24
- Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.
Programs
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Mathematica
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])