A002962 -id:A002962 - cp's OEIS frontend

A006983 Number of simple perfect squared squares of order n up to symmetry.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 26, 160, 441, 1152, 3001, 7901, 20566, 54541, 144161, 378197, 990981, 2578081, 6674067, 17086918

Offset: 1

N. J. A. Sloane

A squared rectangle (which may be a square) is a rectangle dissected into a finite number of two or more squares. If no two 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. - Geoffrey H. Morley, Oct 17 2012

J.-P. Delahaye, Les inattendus mathématiques, Belin-Pour la Science, Paris, 2004, pp. 95-96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stuart E. Anderson, Perfect Squared Rectangles and Squared Squares
Stuart E. Anderson, Simple perfect squared squares in orders 27 to 37 - methods used and people involved.
C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992) 67-75.
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.
C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, Correspondence, 1971.
G. Brinkmann and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007), 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
A. J. W. Duijvestijn, Illustration for a(21)=1 (The unique simple squared square of order 21. Reproduced with permission of the discoverer.)
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24, J. Combin. Theory Ser. B 59 (1993), 26-34.
A. J. W. Duijvestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332. doi:10.1090/S0025-5718-1994-1208220-9
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. doi:10.1090/S0025-5718-96-00705-3 [TableI List of Simple Perfect Squared Squares of order 26 and TableII List of Simple Perfect Squared 2x1 Rectangles of order 26 are now on squaring.net and no longer located as described in the paper.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25.
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Index entries for squared squares

Cf. A002962, A002881, A002839, A014530.
Cf. A181340, A181735, A217155, A217156.
Cf. A129947, A217149, A228953 (related to sizes of the squares).
Cf. A349205, A349206, A349207, A349208, A349209, A349210 (related to ratios of element and square sizes).

Leading term changed from 0 to 1, Apr 15 1996
More terms from Stuart E Anderson, May 08 2003, Nov 2010
Leading term changed back to 0, Dec 25 2010 (cf. A178688)
a(29) added by Stuart E Anderson, Aug 22 2010; contributors to a(29) include Ed Pegg Jr and Stephen Johnson
a(29) changed to 7901, identified a duplicate tiling in order 29. - Stuart E Anderson, Jan 07 2012
a(28) changed to 3000, identified a duplicate tiling in order 28. - Stuart E Anderson, Jan 14 2012
a(28) changed back to 3001 after a complete recount of order 28 SPSS recalculated from c-nets with cleansed data, established the correct total of 3001. - Stuart E Anderson, Jan 24 2012
Definition clarified by Geoffrey H. Morley, Oct 17 2012
a(30) added by Stuart E Anderson, Apr 10 2013
a(31), a(32) added by Stuart E Anderson, Sep 29 2013
a(33), a(34) and a(35) added by Stuart E Anderson, May 02 2016
Moved comments on orders 27 to 35 to a linked file.  Stuart E Anderson, May 02 2016
a(36) and a(37) enumerated by Jim Williams, added by Stuart E Anderson, Jul 26 2020.

A002839 Number of 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, 4931308, 17633773, 63301427, 228130926, 825229110, 2994833854

Offset: 1

N. J. A. Sloane

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 03 2024]

See A217156 for further references and links.
C. J. Bouwkamp, personal communication.
C. J. Bouwkamp, A. J. W. Duijvestijn and P. Medema, Catalogue of simple squared rectangles of orders nine through fourteen and their elements, Technische Hogeschool, Eindhoven, The Netherlands, May 1960, 50 pp.
C. J. Bouwkamp, A. J. W. Duijvestijn and J. Haubrich, Catalogue of simple perfect squared rectangles of orders 9 through 18, Philips Research Laboratories, Eindhoven, The Netherlands, 1964 (unpublished) vols 1-12, 3090 pp.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
M. E. Lines, Think of a Number, Institute of Physics, London, 1990, p. 43.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Tutte, Squaring the Square, in M. Gardner's 'Mathematical Games' column in Scientific American 199, Nov. 1958, pp. 136-142, 166. Reprinted with addendum and bibliography in the US in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, Simon and Schuster, New York (1961), pp. 186-209, 250 [sequence p. 207], and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-7 [sequence p. 162].

S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
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).
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).
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, A. J. W. Duijvestijn and P. Medema, Tables relating to simple squared rectangles of orders nine through fifteen, Technische Hogeschool, Eindhoven, The Netherlands, August 1960, ii + 360 pp. Reprinted in EUT Report 86-WSK-03, January 1986. [Sequence p. i.]
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971.
R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340. Reprinted in I. Gessel and G.-C. Rota (editors), Classic papers in combinatorics, Birkhäuser Boston, 1987, pp. 88-116. [Pp. 324-5 of the original article have counts up to a(12).]
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523-612.
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24. [Number of simple rectangles excludes squares in separate column (from order 21).]
D. Moews, Squared rectangles
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
J. H. van Lint, Letter to N. J. A. Sloane, N.D.
Eric Weisstein's World of Mathematics, Perfect Square Dissection.
Index entries for squared rectangles
Index entries for squared squares

Cf. A006983, A002962, A002881, A181735.

Cf. A217153, A217154, A217156, A219766.

From Stuart E Anderson, Mar 02 2011, Feb 03 2024: (Start)
In "A Census of Planar Maps", p. 267, 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):
Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). (End)
a(n) = A006983(n) + A219766(n). - Stuart E Anderson, Dec 07 2012

Definition corrected to include 'simple'. 'Simple' and 'perfect' defined in comments. - Geoffrey H. Morley, Mar 11 2010
Corrected a(18) and extended terms to order 21.  All 3-connected planar graphs up to 22 edges used to generate dissections. Imperfect squared rectangles, compound squared rectangles, and all squared squares filtered out leaving simple perfect squared rectangles. - Stuart E Anderson, Mar 2011
Corrected a(18) to a(21) after removing last remaining compounds. - Stuart E Anderson, Apr 10 2011
Added a(22), a(23) and a(24) from Ian Gambini's thesis and corrected a(22). Added I. Gambini's thesis reference. - Stuart E Anderson, May 08 2011
Added some additional references, previous correction to a(22) is an increase of 4 based on a new count of order 22. - Stuart E Anderson, Jul 13 2012
Terms a(21)-a(24) corrected to include squares by Geoffrey H. Morley, Oct 17 2012
a(22)=17633773 from Stuart E Anderson confirmed by Geoffrey H. Morley, Nov 28 2012
a(23)-a(24) from Gambini 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

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

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

A002881 Number of simple imperfect squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 34, 104, 283, 953, 3029, 9513, 30359, 98969, 323646, 1080659, 3668432, 12608491, 43745771, 153812801

Offset: 1

N. J. A. Sloane

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. The order of a squared rectangle is the number of its constituent squares. [Geoffrey H. Morley, Oct 17 2012]

C. J. Bouwkamp, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Tutte, Squaring the Square, in M. Gardner's "Mathematical Games" column in Scientific American 199, Nov. 1958, pp. 136-142, 166, Reprinted with addendum and bibliography in the US in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, Simon and Schuster, New York (1961), pp. 186-209, 250 [sequence on p. 207], and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-187 [sequence on p. 162].

S. E. Anderson, Simple Imperfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Squares.
C. J. Bouwkamp, A. J. W. Duijvestijn and P. Medema, Tables relating to simple squared rectangles of orders nine through fifteen, Technische Hogeschool, Eindhoven, The Netherlands, August 1960, ii + 360 pp. Reprinted in EUT Report 86-WSK-03, January 1986. [Sequence p. i.]
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971.
Eric Weisstein's World of Mathematics, Perfect Rectangle.
Index entries for squared rectangles
Index entries for squared squares

Cf. A006983, A002962, A002839, A220165.

Cf. A181735, A217153, A217154, A217156.

a(n) = A002962(n) + A220165(n).

Edited ("simple" added to the definition, definition of "simple" given in the comments), terms a(13), a(15), a(16), a(17), and a(18) corrected, and terms extended to a(20) by Stuart E Anderson, Mar 09 2011
a(16)-a(20) corrected (excess compounds removed) by Stuart E Anderson, Apr 10 2011
Sequence reverted to the one in Bouwkamp et al. (1960), Gardner (1961), Sloane (1973), and Sloane & Plouffe (1995), which includes simple imperfect squares, by Geoffrey H. Morley, Oct 17 2012
a(19)-a(20) corrected, a(21)-a(24) added by Stuart E Anderson, Dec 03 2012
a(25) from Stuart E Anderson, May 07 2024
a(26) from Stuart E Anderson, Jul 28 2024

A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 26 2021

nonn

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.
It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".
In 1983, A. Augusteijn and A. J. W. Duijvestijn  described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")
For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).
In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.
In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

			Examples for n <= 21 are given in the Pfoertner links. Visualizations of tilings corresponding to optimal networks for n <= 12 are given in the Mathworld "Mrs. Perkins's Quilt" link.

Stuart Anderson, Simple Perfect Squared Squares (SPSSs), Order 21 to 37 and higher orders.
Stuart Anderson, SPSS, Order 21.
Stuart Anderson, Simple Perfect Square-Cylinders (SPSCs); Order 20.
Stuart Anderson, Mrs Perkins's Quilt.
A. Augusteijn and A. J. W. Duijvestijn, Simple perfect square-cylinders of low order, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
A. J. W. Duijvestijn, Simple perfect squared square of lowest order, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
Ed Pegg Jr., List of solutions for the Mrs. Perkins's Quilt Square packing problem.
Hugo Pfoertner, Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors, May 2021, Jan-Apr 2023
Hugo Pfoertner, Visualization of the resistor networks with n >= 13 using the Mathematica graph function, Jan-Apr 2023
Rainer Rosenthal, Cascade graphs for examples n <= 21, January 2023
Rainer Rosenthal, Network and semi-quilt visualizing a(18), January 2023
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt.

Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.

a(n) = n for n >= 18.

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

A220165 Number of nonsquare simple imperfect squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 33, 104, 280, 948, 3014, 9494, 30302, 98897, 323372, 1080168, 3666666, 12604812, 43734613, 153788715

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle 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. The order of a squared rectangle is the number of constituent squares.

See A002881 and A006983.

Stuart E. Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24

Cf. A002881, A006983, A002962, A002839.

a(25) from Stuart E Anderson, May 07 2024

a(26) from Stuart E Anderson, Jul 28 2024

A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 21, 30, 29, 20, 25, 30, 12, 19, 24, 17, 13, 13, 18, 14, 19, 14, 15, 15, 15, 20, 15, 20, 16, 22, 16, 16, 17, 21, 22, 15, 13, 16, 18, 14, 14, 14, 17, 15, 11, 10, 12, 13, 4, 11, 8, 9, 7, 11, 4, 9, 8, 8, 8, 6, 8

Offset: 1

Vitor Pimenta dos Reis Arruda, May 15 2020

nonn

See (Gambini, 1999) for a way to construct the sequence. Actually, one would have to extend Gambini's idea by putting extra 1-sided squares in the list of "usable squares" to allow finding nonzero-waste packings.

			For n=5, squares of sides {1, 4} can be packed inside the container, leading to uncovered area a(5) = 5*5 - (4*4 + 1*1) = 8. The other maximal packable set is composed of the squares sided {1,2,3}, which would lead to uncovered area greater than 8.

Vitor Pimenta dos Reis Arruda, Table of n, a(n) for n = 1..101
I. Gambini, A method for cutting squares into distinct squares, Discrete Applied Mathematics, 98 (1999), 65-80.
Vitor Pimenta dos Reis Arruda, Non trivial decompositions until a(101)
Vitor Pimenta dos Reis Arruda, Luiz Gustavo Bizarro Mirisola, and Nei Yoshihiro Soma, Almost squaring the square: optimal packings for non-decomposable squares, Pesqui. Oper. (2022) Vol. 42.
Giovanni Resta, Illustration of terms a(15)-a(31)
Wikipedia, Squaring the square

Cf. A006983, A002962, A181735, A002839, A228953.

Terms a(17)-a(31) from Giovanni Resta, May 15 2020

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

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

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 03 2024]

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

A006983 Number of simple perfect squared squares of order n up to symmetry.

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A002839 Number of simple perfect squared rectangles of order n up to symmetry.

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

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A002881 Number of simple imperfect squared rectangles of order n up to symmetry.

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A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

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

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A220165 Number of nonsquare simple imperfect squared rectangles of order n up to symmetry.

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A334905 a(n) is the minimum remaining space when a square n X n is tiled with smaller squares with distinct integer sides parallel to the n X n square.

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