A340726 -id:A340726 - cp's OEIS frontend

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.

1, 1, 2, 5, 11, 29, 84, 267, 921, 3481, 14322, 62306, 285845, 1362662, 6681508, 33483830

Offset: 1

Kevin Johnston, Feb 11 2016

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.
From Rainer Rosenthal, Dec 23 2022: (Start)
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.)
.
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)

			From _Rainer Rosenthal_, Dec 24 2022, updated May 09 2024: (Start)
.
                                 |A|
     |A B|                       |B|
     |C D|  (2 X 2: 1,1,1,1)     |C|    (4 X 1: 1,1,1,1)
                                 |D|
.
                                 |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|
.
    |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
  ________________________________________________________
       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
.
Example configurations for a(6) = 29:
.
                                    |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)
.                                       _________________________
      |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|      |    6      |             |
      |A A A A A A B B B B B B B|      |           |      7      |
      |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|      |         |1|_____________|
      |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|      |    5    |  4    |  4    |
      |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)           (13 X 11: 7,6,5,4,4,1)
         [rotated by 90 degrees]         [alternate visualization]
.(End)

See A002839 and A217156 for further references and links.

Stuart E. Anderson, Perfect Squared Rectangles and Squared Squares

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

a(15)-a(16) from Kevin Johnston, Feb 11 2016

Title changed from Rainer Rosenthal, Dec 28 2022

A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

Original entry on oeis.org

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536

Offset: 1

Rainer Rosenthal, Nov 12 2020

A219158 gives the minimum number of squares to tile an i x j rectangle. a(n) is found by checking all rectangles (i,j) for which A219158 has a dissection into n squares.

Due to the potential counterexamples to the minimal squaring conjecture (see MathOverflow link), terms after a(19) have to be considered only as lower bounds: a(20) >= 876696755, a(21) >= 2735106696. - Hugo Pfoertner, Nov 17 2020, Apr 02 2021

			a(6) = 11*13 = 143.
Dissection of the 11 X 13 rectangle into 6 squares:
.
          +-----------+-------------+
          |           |             |
          |           |             |
          |   6 X 6   |    7 X 7    |
          |           |             |
          |           |             |
          +---------+-+             |
          |         +-+-----+-------+
          |  5 X 5  |       |       |
          |         | 4 X 4 | 4 X 4 |
          |         |       |       |
          +---------+-------+-------+
.
a(19) = 16976*17061 = 289627536.
Dissection of the 16976 X 17061 rectangle into 19 squares:
.
       +----------------+-------------+
       |                |             |
       |                |             |
       |                |     7849    |
       |      9212      |             |
       |                |             |
       |                |             |
       |                |------+------|
       |________________|      |      |
       |             |   see   | 4109 |
       |             |Rosenthal|      |
       |             |  link +-+------+
       |     7764    |-------|        |
       |             |       |  5018  |
       |             | 4279  |        |
       |             |       |        |
       +-------------+-------+--------+
.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles.
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.
Bertram Felgenhauer, Filling rectangles with integer-sided squares.
MathOverflow, tiling a rectangle with the smallest number of squares, answer by Ed Pegg Jr, Jul 09 2017.
Rainer Rosenthal, Rectangle tiled by 19 squares with maximum area a(19)

Cf. A219158, A340726, A340920.

This sequence and A089047 are effectively analogs for dissecting (or tiling) rectangles and squares respectively. Analogs using equilateral triangular tiles are A014529 and A290821 respectively.

a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020
a(18) from Hugo Pfoertner, Feb 18 2021
a(19) from Hugo Pfoertner, Apr 02 2021

A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 24, 40, 69, 130, 231, 408, 689, 1272, 2153, 3960, 6993, 12560

Offset: 1

Rainer Rosenthal, Jan 16 2021

a(n) is taken from the set of resistance values counted by A337517(n). These sets can be computed by the PARI program of Andrew Howroyd in A180414.
Also the maximum numerator of these electrical networks for small n.
Maximum numerator and maximum denominator coincide for planar networks: for every resistance R in a planar network with n resistors there is always another planar network with n resistors and resistance 1/R. For nonplanar networks this is not necessarily so, as can be seen in A338573.
The asymmetry is illustrated by the example a(15) = 2153.
The author conjectures that this asymmetry will increase with n, and eventually the maxima will differ.
Conjecture: a(19) = 22233, a(20) = 39918. It would be very desirable to know at which value of n > 18 the maximum values of numerators and denominators differ for the first time. - Hugo Pfoertner, Apr 19 2021

			Denominators for numerator a(15) = 2153 in electrical networks with 15 resistors:
  1025,1049,1051,1058,1089,1104,1145,1184,1185,1193,1208,
  1212,1219,1248,1254,1337,1382,1403,1526,1527,1529,1530,
  1545,1547,1555,1579,1586,1632,1642,1647,1687,1699,1719.
Numerators for denominator a(15) = 2153 in electrical networks with 15 resistors:
   899, 905, 934, 941, 945, 960, 968, 969,1008,1049,1064,
  1095,1102,1104,1128,1137,1143,1147,1164,1182,1207,1296,
  1359,1367,1387,1400,1447,1543.

Index to sequences related to resistances.

Cf. A180414, A337517, A338601, A339548, A339808, A340726.

a(18) from Hugo Pfoertner, Apr 11 2021

cp's OEIS Frontend

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.

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A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

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A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

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Author

Keywords

Comments

Examples

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Extensions