A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

1, 1, 21, 269, 4489, 82981, 2995185, 118897973

Offset

1,3

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

Links

Table of n, a(n) for n=1..8.

Example

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) = 21. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |   |       |   |       |   |   |           |
  +           +   +---+---+---+   +---+---+   +   +---+---+---+
  |           |   |           |   |       |   |   |           |
  +           +   +           +   +       +   +   +           +
  |           |   |           |   |       |   |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.

Crossrefs

Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.

Sequence in context: A133105 A215766 A320819 * A228215 A032535 A022745

Adjacent sequences: A360496 A360497 A360498 * A360500 A360501 A360502

Keyword

nonn,more

Author

Scott R. Shannon, Feb 09 2023

Status

approved