A160911 - OEIS

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.

%I #68 May 10 2024 12:29:09

%S 1,1,2,5,11,29,84,267,921,3481,14322,62306,285845,1362662,6681508,

%T 33483830

%N 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 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 Short notation for the 2 possible 3-tile solutions:

%C 3 X 1: 1,1,1

%C 3 X 2: 2,1,1

%C More examples see below.

%C 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 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 From _Rainer Rosenthal_, Dec 23 2022: (Start)

%C The above description can be abbreviated as follows:

%C a(n) is the number of (2+n)-tuples (p X q: t_1,...,t_n) of positive integers, such that:

%C 0. p >= q.

%C 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 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 .

%C 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 (End)

%D See A002839 and A217156 for further references and links.

%H Stuart E. Anderson, <a href="http://www.squaring.net/">Perfect Squared Rectangles and Squared Squares</a>

%e From _Rainer Rosenthal_, Dec 24 2022, updated May 09 2024: (Start)

%e .

%e |A|

%e |A B| |B|

%e |C D| (2 X 2: 1,1,1,1) |C| (4 X 1: 1,1,1,1)

%e |D|

%e .

%e |A A|

%e |A A A| |A A|

%e |A A A| |B B|

%e |A A A| (4 X 3: 3,1,1,1) |B B| (5 X 2: 2,2,1,1)

%e |B C D| |C D|

%e .

%e |A A A|

%e |A A A| <================= 3 X 3 minor A

%e |A A A| 2 X 2 minor B

%e |B B C| (5 X 3: 3,2,1,1) 1 X 1 minor C

%e |B B D| 1 X 1 minor D

%e ________________________________________________________

%e a(4) = 5 illustrated as (p X q: t_1,t_2,t_3,t_4)

%e and as p X q matrices with t_i X t_i minors

%e .

%e Example configurations for a(6) = 29:

%e .

%e |A A A A|

%e |A A B| |A B| |A A A A|

%e |A A C| |C D| |B B C D|

%e |D E F| |E F| |B B E F|

%e ______________________________________________

%e (3 X 3: (3 X 2: (6 X 4:

%e 2,1,1,1,1,1) 1,1,1,1,1,1) 4,2,1,1,1,1)

%e . _________________________

%e |A A A A A A B B B B B B B| | | |

%e |A A A A A A B B B B B B B| | 6 | |

%e |A A A A A A B B B B B B B| | | 7 |

%e |A A A A A A B B B B B B B| | | |

%e |A A A A A A B B B B B B B| |___________| |

%e |C C C C C D B B B B B B B| | |1|_____________|

%e |C C C C C E E E E F F F F| | | | |

%e |C C C C C E E E E F F F F| | 5 | 4 | 4 |

%e |C C C C C E E E E F F F F| | | | |

%e |C C C C C E E E E F F F F| |_________|_______|_______|

%e _____________________________ _____________________________

%e (13 X 11: 7,6,5,4,4,1) (13 X 11: 7,6,5,4,4,1)

%e [rotated by 90 degrees] [alternate visualization]

%e .(End)

%Y Cf. A002839, A005670, A113881, A210517, A217156, A219924, A221843, A221844, A221845, A340726, A342558, A350237.

%K nonn,more

%O 1,3

%A _Kevin Johnston_, Feb 11 2016

%E a(15)-a(16) from _Kevin Johnston_, Feb 11 2016

%E Title changed from _Rainer Rosenthal_, Dec 28 2022