A361001 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling), 1 <= k <= n.

1, 2, 4, 3, 7, 9, 4, 11, 18, 23, 4, 14, 22, 34, 41, 6, 23, 42, 72, 108

Offset: 1

Author

Pontus von Brömssen, Feb 28 2023

Keywords

• nonn
• tabl
• more

Examples

			Triangle begins:
  n\k|  1  2  3  4   5  6
  ---+-------------------
  1  |  1
  2  |  2  4
  3  |  3  7  9
  4  |  4 11 18 23
  5  |  4 14 22 34  41
  6  |  6 23 42 72 108  ?
The T(3,3) = 9 nonrearrangeable tilings of the 3 X 3 square are:
  +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |       |   |
  +           +   +---+---+---+   +---+---+---+
  |           |   |           |   |           |
  +           +   +           +   +           +
  |           |   |           |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |       |   |   |   |   |   |
  +---+---+---+   +---+---+   +   +---+---+   +
  |           |   |       |   |   |       |   |
  +           +   +       +   +   +       +   +
  |           |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |           |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+
  |       |   |   |           |   |   |   |   |
  +       +---+   +---+---+---+   +---+---+---+
  |       |   |   |           |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+
		

Crossrefs

Cf. A000005, A360629, A360998, A361002 (main diagonal), A361003 (first column), A361004 (second column), A361005 (third column).

Formula

T(n,1) = A361003(n) = A000005(n) + floor((n-1)/2). (The first term corresponds to cases where all pieces have the same size, the second to cases where there are two pieces of different sizes.)