A368256 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k cylinder up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflections.

1, 2, 2, 3, 5, 2, 6, 14, 9, 4, 10, 44, 52, 26, 4, 20, 152, 366, 298, 62, 8, 36, 560, 2800, 4244, 1704, 205, 9, 72, 2144, 22028, 66184, 52740, 11228, 623, 18, 136, 8384, 175296, 1050896, 1679776, 701124, 75412, 2171, 23

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     3       6         10           20
   2 | 2   5    14      44        152          560
   3 | 2   9    52     366       2800        22028
   4 | 4  26   298    4244      66184      1050896
   5 | 4  62  1704   52740    1679776     53696936
   6 | 8 205 11228  701124   44758448   2863442960
   7 | 9 623 75412 9591666 1227199056 157073688884
		

Links

• Peter Kagey, Illustration of T(2,3)=14
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Programs

Mathematica

A368256[n_, m_] := 1/(4n)*( DivisorSum[n, EulerPhi[#]*2^(n*m/#) &] + n (2^(n*m/2 - 1))*Boole[EvenQ[n]] + If[EvenQ[m], DivisorSum[n, EulerPhi[#]*2^(n*m/LCM[#, 2]) &], DivisorSum[n, EulerPhi[#]*2^(n*m/#) &, EvenQ]] + n*2^(n*m/2)*Which[EvenQ[m], 1, EvenQ[n], 3/2, True, Sqrt[2]])