A368218 Table read by downward 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 horizontal reflection only.

1, 3, 2, 4, 7, 3, 10, 20, 24, 6, 16, 76, 144, 76, 10, 36, 272, 1120, 1056, 288, 20, 64, 1072, 8448, 16576, 8320, 1072, 36, 136, 4160, 66816, 262656, 263680, 65792, 4224, 72, 256, 16576, 528384, 4197376, 8396800, 4197376, 525312, 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    3     4      10        16          36
    2 |  2    7    20      76       272        1072
    3 |  3   24   144    1120      8448       66816
    4 |  6   76  1056   16576    262656     4197376
    5 | 10  288  8320  263680   8396800   268517376
    6 | 20 1072 65792 4197376 268451840 17180065792
		

Links

• Peter Kagey, Illustration of T(3,2)=24
• 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, A367524, A368219, A368220, A368221.

Programs

Mathematica

A368218[n_, m_] := 2^(n*m/2 - 2)*(2^(n*m/2) + Boole[EvenQ[n*m]] + Boole[EvenQ[m]] + If[EvenQ[n], 1, 2^(m/2)])