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COMMENTS
| There is only one arrangement of 1 square tile: a 1 X 1 rectangle. There is also only 1 arrangement of 2 sq tiles: a 2 X 1 rect. There are 2 arrangements of 3 sq tiles: a 3 X 1 rect (3 1 X 1 tiles) and a 3 X 2 rect (a 2 X 2 tile and 2 1 X 1 tiles).
If we use this notation for the 2 possible 3 tile solutions:
3 X 1:1,1,1
3 X 2:2,1,1
Then the 5 possible 4 tile solutions are:
4 X 1:1,1,1,1
4 X 3:3,1,1,1
5 X 3:3,2,1,1
2 X 2:1,1,1,1
5 X 2:2,2,1,1
The smallest tile is not always a unit tile, e.g. one of the solutions for 5 tiles is:
6 X 5:3,3,2,2,2
My definition of a unique solution is the "signature" string in this notation: the irreducible rectangle size and the list of tile sizes sorted largest to smallest. Rotations and reflections of a known solution are not new solutions, rearrangements of the same size tiles within the same overall boundary are not new solutions. But reorganizations of the same size tiles in different boundaries are unique solutions, such as 4x1:1,1,1,1 and 2x2:1,1,1,1.
I am currently trying to find a(15) and a(16).
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