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

Author

Peter Kagey, Dec 21 2023

Keywords

• nonn
• tabl

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.

Crossrefs

Cf. A222188, A368219, A368256, A368302, A368304, A368307.

Programs

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])