A368306 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k torus up to horizontal reflections by a tile that is not fixed under horizontal reflection.

1, 2, 2, 2, 5, 2, 4, 8, 9, 4, 4, 24, 32, 26, 4, 8, 56, 186, 182, 62, 9, 10, 190, 1096, 2130, 1096, 205, 10, 20, 596, 7356, 26296, 26380, 7356, 623, 22, 30, 2102, 49940, 350316, 671104, 350584, 49940, 2171, 30

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     24       56       190
   3 | 2   9   32    186     1096      7356
   4 | 4  26  182   2130    26296    350316
   5 | 4  62 1096  26380   671104  17899020
   6 | 9 205 7356 350584 17897924 954481360
		

Links

• Peter Kagey, Illustration of T(3,3)=32
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Crossrefs

Cf. A368222, A368259, A368304, A368305, A368308, A184271.

Programs

Mathematica

A368306[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*If[EvenQ[n], DivisorSum[m, EulerPhi[#] (2^(n*m/LCM[2, #]) + 2^((n - 2)*m/LCM[2, #])*4^(m/#)*Boole[EvenQ[#]]) &]/2, DivisorSum[m, EulerPhi[#]*2^(n*m/#) &, EvenQ]])