A360804
Number of ways to tile an n X n square using rectangles with distinct areas.
Original entry on oeis.org
1, 1, 21, 253, 2401, 36237, 815929, 18713197
Offset: 1
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:
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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:
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as this contains two rectangles with area 4. This can occur in 16 different ways so a(4) = A360499(4) - 16 = 269 - 16 = 253.
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.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 360, 0, 360, 360, 8547192, 0
Offset: 1
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:
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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.
Original entry on oeis.org
0, 1, 1, 0, 1, 0, 1, 0, 0, 4128, 1, 10880, 641, 45904, 349496, 892088, 40873, 17695080
Offset: 1
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:
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