A368308 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by a tile that is not fixed under 180-degree rotation.

1, 2, 2, 2, 5, 2, 4, 9, 9, 4, 4, 26, 32, 26, 4, 9, 62, 192, 192, 62, 9, 10, 205, 1096, 2174, 1096, 205, 10, 22, 623, 7440, 26500, 26500, 7440, 623, 22, 30, 2171, 49940, 351336, 671104, 351336, 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         9
   2 | 2   5    9     26       62       205
   3 | 2   9   32    192     1096      7440
   4 | 4  26  192   2174    26500    351336
   5 | 4  62 1096  26500   671104  17904476
   6 | 9 205 7440 351336 17904476 954546880
		

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. A184271, A368224, A368263, A368304, A368306, A368307.

Programs

Mathematica

A368308[n_, m_] := 1/(2*n*m)*(DivisorSum[n, Function[d, DivisorSum[m, EulerPhi[#] EulerPhi[d] 2^(m*n/LCM[#, d]) &]]] + n*m*2^(n*m/2)*Which[OddQ[n*m], 0, OddQ[n + m], 1/2, True, 3/4])