A368307 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k torus up to 180-degree rotation by two tiles that are both fixed under 180-degree rotation.

2, 3, 3, 4, 7, 4, 6, 13, 13, 6, 8, 34, 48, 34, 8, 13, 78, 224, 224, 78, 13, 18, 237, 1224, 2302, 1224, 237, 18, 30, 687, 7696, 27012, 27012, 7696, 687, 30, 46, 2299, 50964, 353384, 675200, 353384, 50964, 2299, 46

Offset: 1

Author

Peter Kagey, Dec 21 2023

Keywords

• nonn
• tabl

Examples

			Table begins:
  n\k|  1   2    3      4        5         6
  ---+--------------------------------------
   1 |  2   3    4      6        8        13
   2 |  3   7   13     34       78       237
   3 |  4  13   48    224     1224      7696
   4 |  6  34  224   2302    27012    353384
   5 |  8  78 1224  27012   675200  17920860
   6 | 13 237 7696 353384 17920860 954677952
		

Links

• Peter Kagey, Illustration of T(3,3)=48
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Crossrefs

Cf. A368223, A368262, A368303, A368308.

Programs

Mathematica

A368307[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], Sqrt[2], OddQ[n + m], 3/2, True, 7/4])