A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

1, 1, 33, 513, 14409, 693025, 50447161

Offset: 1

Author

Scott R. Shannon, Feb 17 2023

Keywords

• nonn
• more

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct height X width dimensions means that, for example, a 1 X 3 rectangle can be used twice, once in a horizontal (1 X 3) and once in a vertical (3 X 1) direction.

Examples

			a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
a(3) = 33. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |   |       |   |   |       |   |       |   |   |           |   |   |       |
  +---+---+---+   +---+---+---+   +---+---+   +   +---+---+---+   +---+---+---+
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +   +       +   +           +   +       +   +   +           +   +       +   +
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 4 different ways, the second in 8 different ways, the third in 8 different ways, the fourth in 4 different ways and the fifth in 8 different ways. There is also the single 3 X 3 rectangle. This gives 33 ways in total.
		

Crossrefs

Cf. A360725 (oblongs), A360499, A360498, A182275, A004003, A099390, A065072.