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A160911
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a(n) is the number of arrangements of n square tiles with coprime sides in a rectangular frame, counting reflected, rotated or rearranged tilings only once.
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2
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1, 1, 2, 5, 11, 29, 84, 267, 921, 3481, 14322, 62306, 285845, 1362662, 6681508, 33483830
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OFFSET
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1,3
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COMMENTS
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There is only one arrangement of 1 square tile: a 1 X 1 rectangle. There is also only 1 arrangement of 2 square tiles: a 2 X 1 rectangle. There are 2 arrangements of 3 square tiles: a 3 X 1 rectangle (three 1 X 1 tiles) and a 3 X 2 rectangle (a 2 X 2 tile and two 1 X 1 tiles).
Short notation for the 2 possible 3-tile solutions:
3 X 1: 1,1,1
3 X 2: 2,1,1
More examples see below.
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 rectangle size for nonsquares and the list of coprime 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 4 X 1: 1,1,1,1 and 2 X 2: 1,1,1,1.
The above description can be abbreviated as follows:
a(n) is the number of (2+n)-tuples (p X q: t_1,...,t_n) of positive integers, such that:
0. p >= q.
1. gcd(t_1,...,t_n) = 1 and t_i >= t_j for i < j and Sum_{i=1..n} t_i^2 = p * q.
2. Any p X q matrix is the disjoint union of contiguous t_i X t_i minors, i = 1..n. (For contiguous minors resp. submatrices see comments in A350237.)
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The rectangle size p X q may have gcd(p,q) > 1, as seen in the examples for 3 X 2 and 6 X 4. Therefore a(n) >= A210517(n) for all n, and a(6) > A210517(6).
(End)
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REFERENCES
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LINKS
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EXAMPLE
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From Rainer Rosenthal, Dec 24 2022, with correction for 3 X 3 Mar 17 2023: (Start)
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|A|
|A B| |B|
|C D| (2 X 2: 1,1,1,1) |C| (4 X 1: 1,1,1,1)
|D|
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|A A|
|A A A| |A A|
|A A A| |B B|
|A A A| (4 X 3: 3,1,1,1) |B B| (5 X 2: 2,2,1,1)
|B C D| |C D|
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|A A A|
|A A A| <================= 3 X 3 minor A
|A A A| 2 X 2 minor B
|B B C| (5 X 3: 3,2,1,1) 1 X 1 minor C
|B B D| 1 X 1 minor D
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a(4) = 5 illustrated as (p X q: t_1,t_2,t_3,t_4)
and as p X q matrices with t_i X t_i minors
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Example configurations for a(6) = 29:
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|A A A A|
|A A A A|
|A A A A|
|A A B| |A B| |A A A A|
|A A C| |C D| |B B C D|
|D E F| |E F| |B B E F|
______________________________________________
(3 X 3: (3 X 2: (6 X 4:
2,1,1,1,1,1) 1,1,1,1,1,1) 4,2,1,1,1,1)
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|A A A A A A B B B B B B B|
|A A A A A A B B B B B B B|
|A A A A A A B B B B B B B|
|A A A A A A B B B B B B B|
|A A A A A A B B B B B B B|
|A A A A A A B B B B B B B|
|C C C C C D B B B B B B B|
|C C C C C E E E E F F F F|
|C C C C C E E E E F F F F|
|C C C C C E E E E F F F F|
|C C C C C E E E E F F F F|
_____________________________
(13 X 11: 7,6,5,4,4,1)
[rotated by 90 degrees]
.(End)
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CROSSREFS
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Cf. A002839, A005670, A113881, A210517, A217156, A219924, A221843, A221844, A221845, A340726, A342558, A350237.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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