A364457 Number A(n,k) of tilings of a k X n rectangle using dominoes and trominoes (of any shape); square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 6, 1, 1, 1, 2, 17, 30, 17, 2, 1, 1, 2, 43, 145, 145, 43, 2, 1, 1, 3, 108, 733, 1352, 733, 108, 3, 1, 1, 4, 280, 3540, 12688, 12688, 3540, 280, 4, 1, 1, 5, 727, 17300, 115958, 226922, 115958, 17300, 727, 5, 1
Offset: 0
Examples
A(3,2) = A(2,3) = 6: .___. .___. .___. .___. .___. .___. | | | |___| | | | |___| | ._| |_. | | | | |___| |_|_| | | | |_| | | |_| |_|_| |___| |___| |_|_| |___| |___| . . Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 0, 1, 1, 1, 2, 2, 3, ... 1, 1, 2, 6, 17, 43, 108, 280, ... 1, 1, 6, 30, 145, 733, 3540, 17300, ... 1, 1, 17, 145, 1352, 12688, 115958, 1075397, ... 1, 2, 43, 733, 12688, 226922, 3927233, 68846551, ... 1, 2, 108, 3540, 115958, 3927233, 128441094, 4263997124, ... 1, 3, 280, 17300, 1075397, 68846551, 4263997124, 267855152858, ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..350
- Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
- Wikipedia, Domino (mathematics)
- Wikipedia, Tromino
Crossrefs
Formula
A(n,k) = A(k,n).
Extensions
Terms n,k>=4 had to be corrected as was pointed out by Martin Fuller and David Radcliffe - Alois P. Heinz, Apr 05 2025