A360256
Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.
Original entry on oeis.org
1, 1, 33, 513, 14409, 693025, 50447161
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) = 33. The possible tilings, excluding those equivalent by symmetry, are:
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The first tiling can occur in 4 different ways, the second in 8 different ways, the third in 8 different ways, the fourth in 4 different ways and the fifth in 8 different ways. There is also the single 3 X 3 rectangle. This gives 33 ways in total.
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.
Original entry on oeis.org
0, 1, 8, 1024, 620448
Offset: 1
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:
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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:
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where (2*1 + 2*3) + (2*5 + 2*3) + (2*2 + 2*1) + (2*2 + 2*2) + (2*8 + 2*5) = 64.
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.
Showing 1-3 of 3 results.
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