A217156 -id:A217156 - cp's OEIS frontend

A110148 Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.

0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 38, 127, 408, 1375, 4783, 16645, 58059, 203808, 722575

Offset: 1

Tanya Khovanova, Feb 18 2007

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. The order of a squared rectangle is the number of constituent squares. [Geoffrey H. Morley, Oct 12 2012]

C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(12) and an incorrect value for a(13) on p. 1178.]
C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625). [Correct terms up to a(13) on p. 1299.]
I. M. Yaglom, How to dissect a square? (in Russian), Nauka, Moscow, 1968. In djvu format (1.7M), also as this pdf (9.5M). [Terms up to a(13) on pp. 26-7.]
Index entries for squared rectangles
Index entries for squared squares

Cf. A217154 (counts symmetries of any subrectangles as distinct).

Cf. A181735, A217153, A217156.

a(n) = A002839(n) + A217152(n) + A217374(n). - Geoffrey H. Morley, Oct 12 2012

a(n) = a(n-1) + A002839(n) + A002839(n-1) + A217152(n) + A217152(n-1). - Geoffrey H. Morley, Oct 12 2012

Definition corrected and a(14)-a(19) added by Geoffrey H. Morley, Oct 12 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

A217148 Smallest possible side length for a perfect squared square of order n; or 0 if no such square exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 112, 110, 110, 120, 147, 212, 180, 201, 221, 201, 215, 185, 233, 218, 225, 253, 237

Offset: 1

Geoffrey H. Morley, Sep 27 2012

It is not known whether this sequence is the same as sequence A129947. It may be that A129947(33) = 246 and A217148(33) = 234. - Geoffrey H. Morley, Jan 10 2013
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. The order of a squared rectangle is the number of constituent squares.
A squared rectangle is simple if it does not contain a smaller squared rectangle.
The upper bounds shown below for 38 and 40-44 are from J. B. Williams. Those for n = 39 and 45-47 are from Gambini's thesis. - Geoffrey H. Morley, Mar 08 2013
======================================
Upper bounds for a(n) for n = 31 to 59
======================================
   |  +0   +1   +2   +3   +4   +5   +6   +7   +8   +9
======================================================
30 |   -    -    -    -    -    -    -    -   352  360
40 |  328  336  360  413  425  543  601  691  550  583
50 |  644  636  584  685  657  631  751  742  780  958
======================================================
The sequence A129947 has identical terms to A217148 (so far), however they are different as A129947 refers to simple perfect squared squares (SPSSs), while A217148 refers to SPSSs and compound perfect squared squares (CPSSs).  The simples and compounds together are referred to as perfect squared squares (PSSs).  So far it has been observed that all the smallest side lengths belong to SPSSs only. - Stuart E Anderson, Oct 27 2020

S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
Stuart E. Anderson, 'Special' Perfect Squared Squares", accessed 2014. - _N. J. A. Sloane_, Mar 30 2014
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, pp. 73-78.
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A129947, A174386, A181735, A217149, A217156.

a(29) from Stuart E Anderson added by Geoffrey H. Morley, Nov 23 2012
a(30) from Stuart E Anderson and Lorenz Milla added by Geoffrey H. Morley, Jun 15 2013
a(31) and a(32) from Lorenz Milla and Stuart E Anderson, Oct 05 2013
For additional terms see the Ed Pegg link, also A006983. - N. J. A. Sloane, Jul 29 2020
a(33) to a(37) from J. B. Williams added by Stuart E Anderson, Oct 27 2020

A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633765, 63301415, 228130900, 825228950, 2994833413

Offset: 1

Stuart E Anderson, Nov 27 2012

A squared rectangle is a rectangle dissected into a finite number of integer-sized squares. If no two of these squares are the same size then the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle or squared square. The order of a squared rectangle is the number of squares into which it is dissected. [Edited by Stuart E Anderson, Feb 02 2024]

See A006983 and A217156 for references.

Stuart E Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
See A006983 and A217156 for further links.

Cf. A002839, A006983, A002962, A002881, A181735.

Cf. A217153, A217154, A217156.

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A002839 = A@002839;
A006983 = A@006983;
a[n_] := A002839[[n]] - A006983[[n]];
a /@ Range[24] (* Jean-François Alcover, Jan 13 2020 *)

a(n) = A002839(n) - A006983(n).
In "A Census of Planar Maps", p. 267, William Tutte gave a conjectured asymptotic formula for the number, a(n) of perfect squared rectangles of order n:
Conjectured: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). [Corrected by Stuart E Anderson, Feb 02 2024]

a(9)-a(24) enumerated by Gambini 1999, confirmed by Stuart E Anderson, Dec 07 2012
a(25) from Stuart E Anderson, May 07 2024
a(26) from Stuart E Anderson, Jul 28 2024

A220164 Number of simple squared squares of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 5, 15, 19, 57, 72, 275, 499, 1778, 3705, 11318, 24525, 65906, 135599, 333938, 687969, 1681759, 3652677

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the squared rectangle is called perfect, otherwise it is imperfect. The order of a squared rectangle is the number of constituent squares. The case in which the squared rectangle is itself a square is called a squared square. The dissection is simple if it contains no smaller squared rectangle, otherwise it is compound. This sequence counts both perfect and imperfect simple squared squares up to symmetry.

See A006983 and A217156.

S. E. Anderson, Simple Perfect Squared Squares
S. E. Anderson, Simple Imperfect Squared Squares
S. E. Anderson, Mrs Perkins's Quilts
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A006983, A002962, A217156, A089046.

a(n) = A006983(n) + A002962(n).

a(13)-a(29) from Stuart E Anderson, Dec 07 2012
Clarified some definitions in comments and added a(30) - Stuart E Anderson, Jun 03 2013
a(31), a(32) added by Stuart E Anderson, Sep 30 2013

A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 6, 22, 76, 246, 848, 2889, 9964, 34440, 119875, 420525, 1482802, 5254679, 18713933, 66968081, 240735712

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of integer sized 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. The order of a squared rectangle is the number of constituent squares. This sequence counts nonsquare simple perfect squared rectangles and nonsquare simple imperfect squared rectangles.

See A006983 and A217156 for further links.

S. E. Anderson, Simple Perfect Squared Rectangles [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Rectangles [Nonsquare rectangles only]

Cf. A219766, A220165, A002839, A002881.

a(9)-a(24) from Stuart E Anderson Dec 07 2012

A220167 Number of simple squared rectangles of order n up to symmetry.

Original entry on oeis.org

3, 6, 22, 76, 247, 848, 2892, 9969, 34455, 119894, 420582, 1482874, 5254954, 18714432, 66969859, 240739417

Offset: 1

Stuart E Anderson, Dec 06 2012

See A006983 and A217156 for references and links.

S. E. Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Perfect Squared Squares.
S. E. Anderson, Simple Imperfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Squares.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Cf. A002839, A002881, A006983, A002962, A220165, A219766.

a(n) = A002839(n) + A002881(n).
a(n) = A006983(n) + A002962(n) + A220165(n) + A219766(n).
Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)), from "A Census of Planar Maps", p. 267, where William Tutte gave a conjectured asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order). [Corrected by Stuart E Anderson, Feb 03 2024]

a(9)-a(24) from Stuart E Anderson, Dec 07 2012

cp's OEIS Frontend

A110148 Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.

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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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A217148 Smallest possible side length for a perfect squared square of order n; or 0 if no such square exists.

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A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry.

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A220164 Number of simple squared squares of order n up to symmetry.

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A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

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A220167 Number of simple squared rectangles of order n up to symmetry.

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