A360629 Triangle read by rows: T(n,k) is the number of sets of integer-sided rectangular pieces that can tile an n X k rectangle, 1 <= k <= n.

1, 2, 4, 3, 10, 21, 5, 22, 73, 192, 7, 44, 190, 703, 2035, 11, 91, 510, 2287, 8581, 27407, 15, 172, 1196, 6738, 30209, 118461, 399618, 22, 326, 2895, 19160, 102092, 462114

Offset: 1

Author

Pontus von Brömssen, Feb 14 2023

Keywords

• nonn
• tabl
• more

Comments

Pieces are free to rotate by 90 degrees, i.e., an r X s piece and an s X r piece are equivalent. See A360451 for the case when the pieces are fixed.

Examples

			Triangle begins:
   n\k|  1   2    3    4     5      6      7
   ---+--------------------------------------
   1  |  1
   2  |  2   4
   3  |  3  10   21
   4  |  5  22   73  192
   5  |  7  44  190  703  2035
   6  | 11  91  510 2287  8581  27407
   7  | 15 172 1196 6738 30209 118461 399618
   ...
T(2,2) = 4, because a 2 X 2 rectangle can be tiled by: one 2 X 2 piece; two 1 X 2 pieces; one 1 X 2 piece and two 1 X 1 pieces; four 1 X 1 pieces.
The T(3,2) = 10 sets of pieces that can tile a 3 X 2 rectangle are shown in the table below. (Each column on the right gives a set of pieces.)
   length X width |  number of pieces
   ---------------+--------------------
        2 X 3     | 1 0 0 0 0 0 0 0 0 0
        2 X 2     | 0 1 1 0 0 0 0 0 0 0
        1 X 3     | 0 0 0 2 1 1 0 0 0 0
        1 X 2     | 0 1 0 0 1 0 3 2 1 0
        1 X 1     | 0 0 2 0 1 3 0 2 4 6
		

Crossrefs

Cf. A000041 (column k=1), A116694, A224697 (square pieces), A360451 (fixed pieces), A360630 (main diagonal), A360631 (column k=2), A360632 (column k=3).

Extensions

T(7,7) and T(8,k) for k = 1..6 added by Robin Visser, May 09 2025