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

1,3

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.

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.

Links

Table of n, a(n) for n=1..5.

Example

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.

Sequence in context: A007117 A291831 A085533 * A122067 A076688 A004808

Adjacent sequences: A360770 A360771 A360772 * A360774 A360775 A360776

Keyword

nonn,more

Author

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

Status

approved