A360804 Number of ways to tile an n X n square using rectangles with distinct areas.

1, 1, 21, 253, 2401, 36237, 815929, 18713197

Offset

1,3

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct areas means that, for example, only one of the two rectangles with area 4, a 2 X 2 or 1 X 4 rectangle, can be used in any tiling.

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 are the same as those given in the examples of A360499(3).
a(4) = 253. And example tiling of the 4 X 4 square is:
.
  +---+---+---+---+
  |   |       |   |
  +---+---+---+   +
  |           |   |
  +           +   +
  |           |   |
  +---+---+---+---+
  |               |
  +---+---+---+---+
.
which contains rectangles with areas 1, 2, 3, 4, 6. The one tiling, excluding symmetrically equivalent arrangements, that is excluded here but allowed in A360499 is:
.
  +---+---+---+---+
  |       |       |
  +       +       +
  |       |       |
  +---+---+       +
  |       |       |
  +---+---+---+---+
  |               |
  +---+---+---+---+
.
as this contains two rectangles with area 4. This can occur in 16 different ways so a(4) = A360499(4) - 16 = 269 - 16 = 253.

Crossrefs

Cf. A360499, A360498, A360725, A360256, A360773, A182275, A004003.

Sequence in context: A165094 A365849 A379170 * A322500 A241632 A323972

Adjacent sequences: A360801 A360802 A360803 * A360805 A360806 A360807

Keyword

nonn,more

Author

Scott R. Shannon, Feb 21 2023

Status

approved