A368220 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 an asymmetric tile.
1, 6, 6, 16, 76, 16, 72, 1056, 1056, 72, 256, 16576, 65536, 16576, 256, 1056, 262656, 4196352, 4196352, 262656, 1056, 4096, 4197376, 268435456, 1073790976, 268435456, 4197376, 4096, 16512, 67117056, 17180000256, 274878431232, 274878431232, 17180000256, 67117056, 16512
Offset: 1
Examples
Table begins:
n\k | 1 2 3 4 5
----+-----------------------------------------------------------
1 | 1 6 16 72 256
2 | 6 76 1056 16576 262656
3 | 16 1056 65536 4196352 268435456
4 | 72 16576 4196352 1073790976 274878431232
5 | 256 262656 268435456 274878431232 281474976710656
6 | 1056 4197376 17180000256 70368756760576 288230376688582656
Links
- Peter Kagey, Illustration of T(2,2)=76
- 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
A368220[n_, m_] := 2^(n*m - 2)*(2^(n*m) + Boole[EvenQ[n*m]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])