A342558 -id:A342558 - cp's OEIS frontend

A014530 List of sizes of squares occurring in lowest order example of a perfect squared square.

2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50

Offset: 1

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does. The order of a squared rectangle is the number of constituent squares. Duijvestijn's perfect square of lowest order (21) is simple. The lowest order of a compound perfect square is 24. [Geoffrey H. Morley, Oct 17 2012]

See the MathWorld link for an explanation of Bouwkamp code. The Bouwkamp code for the squaring is (50,35,27)(8,19)(15,17,11)(6,24)(29,25,9,2)(7,18)(16)(42)(4,37)(33). [Geoffrey H. Morley, Oct 18 2012]

			Example from _Rainer Rosenthal_, Mar 25 2021: (Start)
.
     Terms   | 2  4  6  7  8 9 11 15 16 17 18 19 24 25 27 29 33 35 37 42 50
  -------------------------------------------------------------------------
             | <-- sort selected groups
  -------------------------------------------------------------------------
  (50,35,27) | .  .  .  .  . .  .  .  .  .  .  .  .  . 27  .  . 35  .  . 50
    (8,19)   | .  .  .  .  8 .  .  .  .  .  . 19  .  .     .  .     .  .
  (15,17,11) | .  .  .  .    . 11 15  . 17  .     .  .     .  .     .  .
    (6,24)   | .  .  6  .    .        .     .    24  .     .  .     .  .
  (29,25,9,2)| 2  .     .    9        .     .       25    29  .     .  .
    (7,18)   |    .     7             .    18                 .     .  .
     (16)    |    .                  16                       .     .  .
     (42)    |    .                                           .     . 42
    (4,37)   |    4                                           .    37
     (33)    |                                               33
  _________________________________________________________________________
       Groups of terms selected and sorted for the Bouwkamp piling
.
  The Bouwkamp code says how to pile up the squares in order to tile the square with side length 50 + 35 + 27 = 112. The procedure is beautifully animated in "Eric Weisstein's World of Mathematics" entry - see link.
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences. Academic Press, San Diego, 1995, Fig. M4482.
I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.

Stuart E. Anderson, Catalogues of Perfect Squared Squares
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
A. J. W. Duijvestijn, Simple perfect square of lowest order, J. Combin. Theory Ser. B 25 (1978), 240-243.
Gergely Földvári, Photo of my artwork (2022) depicting the lowest order perfect squared square using 21 distinct colors
N. D. Kazarinoff and R. Weitzenkamp, On the existence of compound perfect squared squares of small order, J. Combin. Theory Ser. B 14 (1973).163-179. [A compound perfect squared square must contain at least 22 subsquares.]
Trinity College Mathematical Society, The Squared Square
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Index entries for squared squares

Cf. A002839, A002962, A002881, A342558 (related by the analogy between square tilings and resistor networks).

'Simple' removed from definition by Geoffrey H. Morley, Oct 17 2012

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.

Original entry on oeis.org

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

A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

Original entry on oeis.org

1, 3, 5, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21

Offset: 2

Hugo Pfoertner and Rainer Rosenthal, Feb 12 2023

			a(2) = 1, [[1,2]]
.
  1           2
  O----R1R----O
  R_12 = 1
.
a(3) = 3, [[1,2]^2,[2,3]]
.
  1   .---R1R---.   2           3
  O --|         |-- O ---R3R--- O
      .---R2R---.
.
  R_12 = 1/2, R_13 = 3/2,
              R_23 = 1
.
a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]
.
  1           2   .---R2R---.   3          (5)          4
  O ---R1R--- O --|         |-- O ---R4R--- O ---R5R--- O
                  .---R3R---.
.
  R_12 = 1, R_13 = 3/2, R_14 = 7/2,
            R_23 = 1/2, R_24 = 5/2,
                        R_34 = 2
.
a(5) = 8, node 6 hidden,
  [[1, 2], [1, 3]^2, [2, 3], [2, 4], [3, 6], [4, 5], [4, 6]]
.
    1             2           4           5
    O-----R1R-----O----R5R----O----R8R----O
    |             |           |
    |            R4R         R7R
    .---R2R---.   |           |
    |         |---O----R6R----O
    .---R3R---.   3          (6)
.
   R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,
               R_23 = 1/2,  R_24 = 13/18, R_25 = 31/18,
                            R_34 =  8/9,  R_35 = 17/9,
                                          R_45 =  1

IBM Research, Electric networks in graphs, Ponder This Challenge, March 2025, asked for the only network corresponding to a(10)=15 and 4 networks for a(12)=18.
Hugo Pfoertner, Illustrated examples for the terms a(6), a(7), a(8), 17 Feb 2023.
Hugo Pfoertner, Illustrated examples for the terms a(9), a(10), a(11), 3 Apr 2023.
Hugo Pfoertner, Illustration of a(12)=18, 8 Jan 2024, showing 3 planar and 5 non-planar networks, 4 of which were required to solve the bonus question of IBM's Ponder This Challenge.
Hugo Pfoertner and Klaus Nagel, Illustration of a(14)=21, 21 Aug 2025.

Cf. A219158, A342557, A342558, A348020.

a(14) from Klaus Nagel and Hugo Pfoertner, Aug 21 2025

A347282 a(n) is the least number of unit resistors in an electrical network with total resistance A007305(n)/A007306(n) such that all currents through the resistors are distinct and > 0.

Original entry on oeis.org

18, 19, 20, 18, 26, 19, 17, 19

Offset: 1

Hugo Pfoertner, Oct 12 2021

The trivial case of a single resistor is excluded, thus making a(1) = A342558(18) = 18.

			  n  A007305(n)/ a(n)   Edge list of network graph (example)
     A007306(n)
  1     1/1       18    [[1, 5], [1, 6], [1, 7], [2, 5], [2, 7], [2, 8],
                         [3, 6], [3, 7], [3, 8], [3, 9], [10, 6], [4, 8],
                         [4, 9], [5, 8], [5, 9], [6, 9], [7, 9], [10, 4]]
                        Poles: [5, 10]
  2     1/2       19    see linked file for description of solutions
  3     1/3       20
  4     2/3       18
  5     1/4       26
  6     2/5       19
  7     3/5       17
  8     3/4       19
  9     1/5    unknown

Hugo Pfoertner, Examples of networks corresponding to a(1)-a(8), October 2021.
Index to sequences related to resistances.

Cf. A007305, A007306, A342558, A348020.

cp's OEIS Frontend

A014530 List of sizes of squares occurring in lowest order example of a perfect squared square.

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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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A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

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A347282 a(n) is the least number of unit resistors in an electrical network with total resistance A007305(n)/A007306(n) such that all currents through the resistors are distinct and > 0.

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A360031 a(n) is the number of unlabeled 2-connected graphs with n edges containing at least one pair of nodes with resistance distance 1 when all edges are replaced by unit resistors.

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