A360725 Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.

0, 0, 4, 36, 1056, 31052, 1473944, 87469884

Offset: 1

Author

Scott R. Shannon, Feb 18 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 oblong can be used twice, once in a horizonal (1 x 3) and once in a vertical (3 x 1) direction.

Examples

			a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
  +---+---+---+
  |           |
  +---+---+---+
  |           |
  +           +
  |           |
  +---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 36. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |   |   |           |   |   |           |
  +   +           +   +---+---+---+---+   +   +---+---+---+   +   +---+---+---+
  |   |           |   |               |   |   |           |   |   |   |       |
  +---+---+---+---+   +               +   +   +           +   +   +   +       +
  |               |   |               |   |   |           |   |   |   |       |
  +               +   +               +   +---+---+---+---+   +---+---+       +
  |               |   |               |   |               |   |       |       |
  +---+---+---+---+   +---+---+---+---+   +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different ways, the second in 4 different ways, the third in 16 different ways and the fourth in 8 different ways. This gives 36 ways in total.
		

Crossrefs

Cf. A360256 (rectangles), A360499, A360498, A182275, A004003, A099390, A065072.