A360773 Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

0, 1, 8, 1024, 620448

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Feb 20 2023

Keywords

• nonn
• more

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.

Only squares with even edges lengths are possible since the area of a square with odd edge lengths is odd, while the perimeter of any rectangle is even.

Examples

			a(1) = 0 as a 2 x 2 square, with area 4, cannot be tiled with distinct rectangles with perimeters that sum to 4.
a(2) = 1 as a 4 x 4 rectangle, with area 16, can be tiled with a 4 x 4 square with perimeter 4 + 4 + 4 + 4 = 16.
a(3) = 8. The possible tilings for the 6 x 6 square, with area 36, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
  |                       |   |                       |
  +---+---+---+---+---+---+   +                       +
  |                       |   |                       |
  +                       +   +---+---+---+---+---+---+
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
.
where for the first tiling (2*6 + 2*1) + (2*6 + 2*5) = 36 while for the second tiling (2*6 + 2*2) + (2*6 + 2*4) = 36. Both of these tilings can occur in 4 ways, giving 8 ways in total.
a(4) = 1024. And example tiling of the 8 x 8 square, with area 64, is:
.
  +---+---+---+---+---+---+---+---+
  |   |                   |       |
  +   +                   +---+---+
  |   |                   |       |
  +   +                   +       +
  |   |                   |       |
  +---+---+---+---+---+---+---+---+
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +---+---+---+---+---+---+---+---+
.
where (2*1 + 2*3) + (2*5 + 2*3) + (2*2 + 2*1) + (2*2 + 2*2) + (2*8 + 2*5) = 64.
		

Crossrefs

Cf. A360499, A360498, A360725, A360256, A182275, A004003, A099390, A065072.