This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A160911 #68 May 10 2024 12:29:09
%S A160911 1,1,2,5,11,29,84,267,921,3481,14322,62306,285845,1362662,6681508,
%T A160911 33483830
%N A160911 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.
%C A160911 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).
%C A160911 Short notation for the 2 possible 3-tile solutions:
%C A160911 3 X 1: 1,1,1
%C A160911 3 X 2: 2,1,1
%C A160911 More examples see below.
%C A160911 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.
%C A160911 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.
%C A160911 From _Rainer Rosenthal_, Dec 23 2022: (Start)
%C A160911 The above description can be abbreviated as follows:
%C A160911 a(n) is the number of (2+n)-tuples (p X q: t_1,...,t_n) of positive integers, such that:
%C A160911 0. p >= q.
%C A160911 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.
%C A160911 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.)
%C A160911 .
%C A160911 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).
%C A160911 (End)
%D A160911 See A002839 and A217156 for further references and links.
%H A160911 Stuart E. Anderson, <a href="http://www.squaring.net/">Perfect Squared Rectangles and Squared Squares</a>
%e A160911 From _Rainer Rosenthal_, Dec 24 2022, updated May 09 2024: (Start)
%e A160911 .
%e A160911 |A|
%e A160911 |A B| |B|
%e A160911 |C D| (2 X 2: 1,1,1,1) |C| (4 X 1: 1,1,1,1)
%e A160911 |D|
%e A160911 .
%e A160911 |A A|
%e A160911 |A A A| |A A|
%e A160911 |A A A| |B B|
%e A160911 |A A A| (4 X 3: 3,1,1,1) |B B| (5 X 2: 2,2,1,1)
%e A160911 |B C D| |C D|
%e A160911 .
%e A160911 |A A A|
%e A160911 |A A A| <================= 3 X 3 minor A
%e A160911 |A A A| 2 X 2 minor B
%e A160911 |B B C| (5 X 3: 3,2,1,1) 1 X 1 minor C
%e A160911 |B B D| 1 X 1 minor D
%e A160911 ________________________________________________________
%e A160911 a(4) = 5 illustrated as (p X q: t_1,t_2,t_3,t_4)
%e A160911 and as p X q matrices with t_i X t_i minors
%e A160911 .
%e A160911 Example configurations for a(6) = 29:
%e A160911 .
%e A160911 |A A A A|
%e A160911 |A A A A|
%e A160911 |A A A A|
%e A160911 |A A B| |A B| |A A A A|
%e A160911 |A A C| |C D| |B B C D|
%e A160911 |D E F| |E F| |B B E F|
%e A160911 ______________________________________________
%e A160911 (3 X 3: (3 X 2: (6 X 4:
%e A160911 2,1,1,1,1,1) 1,1,1,1,1,1) 4,2,1,1,1,1)
%e A160911 . _________________________
%e A160911 |A A A A A A B B B B B B B| | | |
%e A160911 |A A A A A A B B B B B B B| | | |
%e A160911 |A A A A A A B B B B B B B| | 6 | |
%e A160911 |A A A A A A B B B B B B B| | | 7 |
%e A160911 |A A A A A A B B B B B B B| | | |
%e A160911 |A A A A A A B B B B B B B| |___________| |
%e A160911 |C C C C C D B B B B B B B| | |1|_____________|
%e A160911 |C C C C C E E E E F F F F| | | | |
%e A160911 |C C C C C E E E E F F F F| | 5 | 4 | 4 |
%e A160911 |C C C C C E E E E F F F F| | | | |
%e A160911 |C C C C C E E E E F F F F| |_________|_______|_______|
%e A160911 _____________________________ _____________________________
%e A160911 (13 X 11: 7,6,5,4,4,1) (13 X 11: 7,6,5,4,4,1)
%e A160911 [rotated by 90 degrees] [alternate visualization]
%e A160911 .(End)
%Y A160911 Cf. A002839, A005670, A113881, A210517, A217156, A219924, A221843, A221844, A221845, A340726, A342558, A350237.
%K A160911 nonn,more
%O A160911 1,3
%A A160911 _Kevin Johnston_, Feb 11 2016
%E A160911 a(15)-a(16) from _Kevin Johnston_, Feb 11 2016
%E A160911 Title changed from _Rainer Rosenthal_, Dec 28 2022