A361413 Number of ways to tile an n X n square using rectangles with distinct dimensions where all the rectangle edge lengths are prime numbers.

0, 1, 1, 0, 1, 0, 1, 0, 0, 4128, 1, 10880, 641, 45904, 349496, 892088, 40873, 17695080

Offset: 1

Author

Scott R. Shannon, Mar 10 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 whether it lies horizontally or vertically.

Examples

			a(2), a(3), a(5), a(7), a(11) = 1 as the only possible tiling is that using an n X n square where n is a prime number. It is likely 11 is the last prime indexed term that equals 1 although this is unknown.
a(10) = 4128. And example tiling is:
.
  +---+---+---+---+---+---+---+---+---+---+
  |       |           |                   |
  +       +           +                   +
  |       |           |                   |
  +---+---+---+---+---+---+---+---+---+---+
  |           |                           |
  +           +                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +---+---+---+                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +           +---+---+---+---+---+---+---+
  |           |                           |
  +           +                           +
  |           |                           |
  +           +                           +
  |           |                           |
  +---+---+---+---+---+---+---+---+---+---+
.
		

Crossrefs

Cf. A360943, A360499, A360804, A360256, A360773, A182275, A004003.