A278657 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape and monominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 2, 25, 50, 25, 2, 1, 1, 3, 50, 311, 311, 50, 3, 1, 1, 4, 155, 1954, 4101, 1954, 155, 4, 1, 1, 5, 508, 11914, 56864, 56864, 11914, 508, 5, 1, 1, 6, 1343, 76003, 728857, 1532496, 728857, 76003, 1343, 6, 1
Offset: 0
Examples
A(2,3) = A(3,2) = 7: .___. .___. .___. .___. .___. .___. .___. |_|_| | | | | | |_| |_| | | ._| |_. | |_|_| | ._| |_. | | | | | | |_| |_| | |_|_| |_|_| |_|_| |___| |___| |___| |___| . . Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 2, 3, ... 1, 1, 1, 7, 25, 50, 155, ... 1, 1, 7, 50, 311, 1954, 11914, ... 1, 1, 25, 311, 4101, 56864, 728857, ... 1, 2, 50, 1954, 56864, 1532496, 42238426, ... 1, 3, 155, 11914, 728857, 42238426, 2492016728, ...
Links
- Liang Kai, Antidiagonals n = 0..23, flattened (Antidiagonals n = 0..15 from Alois P. Heinz)
- Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
- Wikipedia, Pentomino