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

1,1

Links

Table of n, a(n) for n=1..45.

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.

Example

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

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])

Crossrefs

Cf. A368223, A368262, A368303, A368308.

Sequence in context: A227385 A049790 A222188 * A368305 A184271 A269098

Adjacent sequences: A368304 A368305 A368306 * A368308 A368309 A368310

Keyword

nonn,tabl

Author

Peter Kagey, Dec 21 2023

Status

approved