A006983 -id:A006983 - cp's OEIS frontend

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

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

A228953 The largest possible element size for each perfect squared square order, otherwise 0 if perfect squared squares do not exist in that order.

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, 50, 97, 134, 200, 343, 440, 590, 797, 1045, 1435, 1855, 2505, 3296, 4528, 5751, 7739, 10361

Offset: 1

Stuart E Anderson, Oct 06 2013

nonn

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. Every perfect square with the largest known element for each order up to 37 is simple.

S. E. Anderson, Perfect Squared Rectangles and Squared Squares
Stuart Anderson, 'Special' Perfect Squared Squares", accessed 2014. - _N. J. A. Sloane_, Mar 30 2014
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Jim Williams, Jim Williams' squared square research.

Cf. A217149, A129947, A006983.

More terms, a(33) to a(37), extracted from Jim Williams' discoveries, added by Stuart E Anderson, Nov 06 2020

A217155 Number of compound perfect squared squares of order n up to symmetries of the square.

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, 0, 0, 0, 4, 12, 100, 220, 948, 2308, 5668, 17351, 52196, 150669, 429458, 1206181, 3337989, 8961794, 23989218, 62894424

Offset: 1

Geoffrey H. Morley, Sep 27 2012

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 compound if it contains a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares.
The terms up to a(26) were first published by Gambini (1999) but included no new squarings neither counted by Duijvestijn, Federico and Leeuw (1982) nor in Skinner's book (1993). In 2010 Anderson and Pegg used plantri and Anderson's programs to confirm Gambini's counts and to find a(27) and a(28).
In 2011, S. E. Anderson and Stephen Johnson commenced order 29 CPSSs, and processed all plantri generated 2-connected minimum degree 3 planar graph embeddings with up to 15 vertices. This left the largest graph class, the 16 vertex class.  In 2012, S. E. Anderson processed the remaining graphs, using the Amazon Elastic Cloud supercomputer and new software which he wrote to find a(29). - Stuart E Anderson, Nov 30 2012
In May 2013, Lorenz Milla and Stuart Anderson enumerated a(30) (CPSSs of order 30), using the same process and software as used on order 29 CPSSs, with the addition of a technique recommended by William Tutte in his writings which resulted in a 3x speed up of the search for perfect squared squares by factoring the determinant of the Kirchhoff/discrete Laplacian matrix of a graph into a product 2fS, where f is a squarefree number and S is a square number. - Stuart E Anderson, May 26 2013
From June to September 2013, Lorenz Milla further optimized the process and software and completed the computation required to enumerate all CPSSs of order 31 and 32.  A second run with enhanced software was undertaken by Milla and Anderson as there was a possibility some CPSSs could have been missed on the first run.  The second run found nothing new or different and confirmed the result. - Stuart E Anderson Sep 29 2013
In April 2014, Jim Williams wrote software and enumerated all CPSSs in orders 33, 34, 35 and 36. - Stuart E Anderson May 02 2016
In August 2018, Jim Williams completed the enumeration of all CPSSs and CPSS isomers in orders 37, 38 and 39. - Stuart E Anderson, Sep 17 2018

			See MathWorld link for an explanation of Bouwkamp code.
a(24)=4 because the compound perfect squares of order 24 comprise the one with side 175 and Bouwkamp code (81,56,38) (18,20) (55,16,3) (1,5,14) (4) (9) (39) (51,30) (29,31,64) (43,8) (35,2) (33) and three others from the other symmetries of the squared subrectangle.

J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993. [Includes some compound perfect squares up to order 30.]

S. E. Anderson, Compound Perfect Squared Squares (complete to order 36).
S. E. Anderson, Compound Perfect Squared Squares of the Order Twenties, arXiv:1303.0599 [math.CO], 2013.
A. J. W. Duijvestijn, P. J. Federico and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 15-32. [The lowest order of a compound perfect square is 24.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25.
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A181340 (counts symmetries of squared subrectangles as equivalent).

Cf. A006983, A217152, A217153, A217156.

a(29) from Stuart E Anderson, Nov 30 2012
a(30) from Stuart E Anderson, May 26 2013
a(31)-a(32) from Stuart E Anderson, Sep 29 2013
Minor edits by Jon E. Schoenfield, Feb 15 2014
a(33)-a(36) from Stuart E Anderson, May 02 2016
a(37)-a(39) from Stuart E Anderson, Sep 17 2018

A349206 a(n) is the side length of the simple perfect squared square of order n leading to a maximum of the ratio of the side length of its smallest element A349205(n) to its total side length.

Original entry on oeis.org

112, 192, 140, 120, 381, 544, 1032, 732, 1615, 1485, 1408, 3584, 1625, 6808, 6192, 7743, 16581

Offset: 21

Hugo Pfoertner, Nov 22 2021

			See A349205.

Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

Cf. A006983, A129947, A217149, A228953, A349205, A349207, A349208, A349209, A349210.

A349209 a(n) is the maximum of the side lengths of the smallest elements of all simple perfect squared squares of order n.

Original entry on oeis.org

2, 4, 2, 8, 16, 20, 48, 48, 69, 74, 107, 158, 177, 270, 355, 519, 716

Offset: 21

Hugo Pfoertner, Nov 17 2021

			See linked illustrations.

Stuart E. Anderson, Catalogues of Perfect Squared Squares.
Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

Cf. A006983, A129947, A217149, A228953, A349205, A349206, A349207, A349208, A349210.

A349210 a(n) is the minimum of the side lengths of the largest elements of all simple perfect squared squares of order n.

Original entry on oeis.org

50, 55, 44, 47, 74, 78, 81, 77, 99, 77, 87, 80, 94, 79, 89, 96, 93

Offset: 21

Hugo Pfoertner, Nov 17 2021

			See linked illustrations.

Stuart E. Anderson, Catalogues of Perfect Squared Squares.
Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

Cf. A006983, A129947, A217149, A228953, A349205, A349206, A349207, A349208, A349209.

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.

A349208 a(n) is the largest side length (size) of an element of the simple perfect squared square of order n leading to a maximum of the ratio of the size of its smallest element A349207(n) to the size of its largest element.

Original entry on oeis.org

50, 86, 60, 47, 195, 202, 457, 304, 591, 698, 520, 769, 549, 860, 3276, 2456, 4098

Offset: 21

Hugo Pfoertner, Nov 22 2021

			See A349207.

Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

Cf. A006983, A129947, A217149, A228953, A349205, A349206, A349207, A349209, A349210.

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

cp's OEIS Frontend

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

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A228953 The largest possible element size for each perfect squared square order, otherwise 0 if perfect squared squares do not exist in that order.

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A217155 Number of compound perfect squared squares of order n up to symmetries of the square.

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A349206 a(n) is the side length of the simple perfect squared square of order n leading to a maximum of the ratio of the side length of its smallest element A349205(n) to its total side length.

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A349209 a(n) is the maximum of the side lengths of the smallest elements of all simple perfect squared squares of order n.

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A349210 a(n) is the minimum of the side lengths of the largest elements of all simple perfect squared squares of order n.

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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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A349208 a(n) is the largest side length (size) of an element of the simple perfect squared square of order n leading to a maximum of the ratio of the size of its smallest element A349207(n) to the size of its largest element.

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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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