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
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.
Programs
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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]])