A360943 Number of ways to tile an n X n square using rectangles with distinct dimensions where no rectangle has an edge length that divides n.

0, 0, 0, 0, 0, 0, 360, 0, 360, 360, 8547192, 0

Offset: 1

Author

Scott R. Shannon, Mar 01 2023

Keywords

• nonn
• more

Comments

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 2 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

Other known values are a(14) = 517344, a(15) = 6068760, a(16) = 339312. a(13) is greater than 800 million.

Examples

			a(1)..a(6),a(8),a(12) = 0 as these squares cannot be tiled with distinct rectangles with edge lengths that do not divide n. For example for the 8 x 8 square only three rectangles are available with dimensions 3 x 3, 3 x 5, and 5 x 5. All other rectangles have an edge length that divides 8 else leave a space of size 1 or 2 units between its edge and the edge of the square. These gaps cannot be filled as no rectangle can have an edge length of 1 or 2.
a(7) = 360. And example tiling is:
.
  +---+---+---+---+---+---+---+
  |       |           |       |
  +       +           +       +
  |       |           |       |
  +---+---+---+---+---+       +
  |                   |       |
  +                   +       +
  |                   |       |
  +---+---+---+---+---+---+---+
  |           |               |
  +           +               +
  |           |               |
  +           +               +
  |           |               |
  +---+---+---+---+---+---+---+
.
		

Crossrefs

Cf. A360499, A360804, A360256, A360773, A182275, A004003.