A368219 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by a tile that is fixed under 180-degree rotation but not horizontal or vertical reflection.

1, 2, 2, 3, 7, 3, 6, 20, 20, 6, 10, 76, 136, 76, 10, 20, 272, 1056, 1056, 272, 20, 36, 1072, 8256, 16576, 8256, 1072, 36, 72, 4160, 65792, 262656, 262656, 65792, 4160, 72, 136, 16576, 524800, 4197376, 8390656, 4197376, 524800, 16576, 136

Offset: 1

Author

Peter Kagey, Dec 18 2023

Keywords

• nonn
• tabl

Examples

			Table begins:
  n\k |  1    2     3       4         5           6
  ----+--------------------------------------------
    1 |  1    2     3       6        10          20
    2 |  2    7    20      76       272        1072
    3 |  3   20   136    1056      8256       65792
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  272  8256  262656   8390656   268451840
    6 | 20 1072 65792 4197376 268451840 17180065792
		

Links

• Peter Kagey, Illustration of T(3,2)=20
• Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-1, A-3.

Crossrefs

Cf. A225910, A368218, A368220, A368223.

Programs

Mathematica

A368219[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + If[EvenQ[n*m], 1, Sqrt[2]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])