A181735 -id:A181735 - 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

A217156 Number of 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, 1, 8, 12, 30, 172, 541, 1372, 3949, 10209, 26234, 71892, 196357, 528866, 1420439, 3784262, 10012056, 26048712

Offset: 1

Geoffrey H. Morley, Sep 27 2012

a(n) is the number of solutions to the classic problem of 'squaring the square' by n unequal squares. 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, and compound if it does.

			a(21) = 1 because there is a unique perfect squared square of order 21. A014530 gives the sizes of its constituent squares.

H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991, section C2, pp. 81-83.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
P. J. Federico, Squaring rectangles and squares: A historical review with annotated bibliography, in Graph Theory and Related Topics, J. A. Bondy and U. S. R. Murty, eds., Academic Press, 1979, 173-196.
J. H. van Lint and R. M. Wilson, A course in combinatorics, Chapter 34 "Electrical networks and squared squares", pp. 449-460, Cambridge Univ. Press, 1992.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.
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, and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-7.
W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 1-11, Clarendon Press, Oxford, 1998.

S. E. Anderson, Simple Perfect Squared Squares (complete to order 29).
S. E. Anderson, Compound Perfect Squared Squares (complete to order 29).
J. A. Bondy and U. S. R. Murty, Chapter 12: The Cycle Space and Bond Space, pp. 212-226 in: Graph theory with applications, Elsevier Science Ltd/North-Holland, 1976.
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.
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]
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.
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523-612.
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.]
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.
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. [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.
C. A. B. Smith and W. T. Tutte, A class of self-dual maps, Can. J. Math., 2 (1950), 179-196.
W. T. Tutte, Squaring the square, Can. J. Math., 2 (1950), 197-209.
W. T. Tutte, The quest of the perfect square, Amer. Math. Monthly 72 (1965), 29-35.
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Wikipedia, Squaring the square
Index entries for squared squares

Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).

Cf. A110148, A217154.

a(n) = A006983(n) + A217155(n).

Added a(29) = 10209, Stuart E Anderson, Nov 30 2012
Added a(30) = 26234, Stuart E Anderson, May 26 2013
Added a(31) = 71892, a(32) = 196357, Stuart E Anderson, Sep 30 2013
Added a(33) = 528866, a(34) = 1420439, a(35) = 3784262, due to enumeration completed by Jim Williams in 2014 and 2016. Stuart E Anderson, May 02 2016
a(36) and a(37) completed by Jim Williams in 2016 to 2018, added by Stuart E Anderson, Oct 28 2020

A129947 Smallest possible side length for a simple 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, 223, 218, 225, 253, 237

Offset: 1

Alexander Adamchuk, Jun 09 2007, corrected Jun 11 2007

It is not known whether this sequence is the same as sequence A217148. It may be that A129947(33) = 246 and A217148(33) = 234. - Geoffrey H. Morley, Jan 10 2013
From Geoffrey H. Morley, Oct 17 2012: (Start)
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 smallest known sides of simple perfect squared squares (and the known orders of the squares) are 110 (22, 23), 112 (21), 120 (24), 139 (22, 23), 140 (23), 145 (23), 147 (22, 25) ...
The upper bounds shown below for n = 38 and 40-44 are from J. B. Williams. The rest are from Gambini's thesis. - Geoffrey H. Morley, Mar 08 2013
======================================
Upper bounds for a(n) for n = 38 to 59
======================================
   |  +0   +1   +2   +3   +4   +5   +6   +7   +8   +9
======================================================
30 |   -    -    -    -    -    -    -   -    352  360
40 |  328  336  360  413  425  543  601  691  621  779
50 |  788  853    ?  824  971  939  929  985 1100 1060
======================================================
(End)

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
Index entries for squared squares

Cf. A006983, A174386, A181735, A217148, A217149, A217156.

Unproved statement misattributed to Skinner replaced, known upper bounds corrected, and crossref added by Geoffrey H. Morley, Mar 19 2010
Additional term added, initial term a(0)=1 deleted by Stuart E Anderson, Dec 26 2010
Upper bounds for terms a(31) to a(78), (all from Ian Gambini's thesis) added by Stuart E Anderson, Jan 20 2011
New bound for a(29)<=221, from Stuart E Anderson & Ed Pegg Jr, Jan 20 2011
a(29) confirmed as 221, from Stuart E Anderson, Ed Pegg Jr, and Stephen Johnson, Aug 22 2011
New bound for a(31)<=236, computed by Stephen Johnson in September 2011, updated by Stuart E Anderson, Oct 04 2011
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

A002962 Number of simple imperfect 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, 274, 491, 1766, 3679, 11158, 24086, 64754, 132598, 326042, 667403, 1627218, 3508516

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 constituent squares. - Geoffrey H. Morley, Oct 17 2012

Orders 15 to 19 were enumerated by C. J. Bowkamp and A. J. W. Duijvestijn (1962). Orders 20 to 29 were enumerated by Stuart Anderson (2010-2012). Orders 30 to 32 were enumerated by Lorenz Milla and Stuart Anderson (2013). - Stuart E Anderson, Sep 30 2013

C. J. Bouwkamp, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stuart Anderson, Simple Imperfect Squared Squares.
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep., 17 (1962), 523-612. [Pp. 573-4 have simple imperfect squares up to order 19.]
Index entries for squared squares

Cf. A002839, A002881, A006983, A014530.

Cf. A181735, A217155, A217156.

a(19) corrected and terms extended up to a(22) by Stuart E Anderson, Mar 08 2011
a(21) and a(22) corrected and terms extended to a(25) by Stuart E Anderson, Apr 24 2011
a(21), a(22), a(25) corrected and a(26)-a(28) added by Stuart E Anderson, Jul 11 2011
a(29) from Stuart E Anderson, Ed Pegg Jr, Stephen Johnson, Aug 22 2011
a(29) corrected by Stuart E Anderson, Aug 24 2011
Definition clarified and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012
a(28) corrected by Stuart E Anderson, Dec 01 2012
a(30) from Lorenz Milla and Stuart E Anderson, Apr 10 2013
a(26) and a(29) corrected by Stuart E Anderson, Aug 20 2013
a(31), a(32) from Lorenz Milla and Stuart E Anderson, Sep 30 2013

A181340 Number of compound perfect squared squares of order n up to symmetries of the square and its squared subrectangles.

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, 1, 2, 16, 46, 143, 412, 941, 2788, 7941, 22413, 62273, 172330, 466508, 1239742, 3257378, 8430928

Offset: 1

Stuart E Anderson, Oct 13 2010, Oct 16 2010

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. - Geoffrey H. Morley, Oct 17 2012
The smallest perfect compound squared square was published by T. H. Willcocks in 1948, has 24 squares and has one rectangle as a sub-dissection; however, it was not until 1982 that A. J. W. Duijvestijn, P. J. Federico and P. Leeuw proved it to be the lowest-order example.
In 2010 Stuart Anderson and Ed Pegg Jr generated all 2-connected minimum degree 3 planar graphs up and including 29 edges, using B. D. McKay and G. Brinkmann's plantri software, then applied electrical node analysis to the graphs to obtain complete counts of compound perfect squares in orders 24, 25, 26, 27 and 28, along with all the members of each equivalence class of each compound square.
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 used it to complete the enumeration of CPSS orders 33, 34, 35 and 36. - Stuart E Anderson, May 02 2016
In August 2018, Jim Williams completed the enumeration of CPSS orders 37, 38 and 39. - Stuart E Anderson, Sep 17 2018.

			From _Geoffrey H. Morley_, Oct 17 2012 (Start):
See MathWorld link for an explanation of Bouwkamp code.
a(24)=1 because all four compound perfect squares of order 24 are equivalent up to symmetries. They have side 175. The Bouwkamp code for one of them is (81,56,38)(18,20)(55,16,3)(1,5,14)(4)(9)(39)(51,30)(29,31,64)(43,8)(35,2)(33). (End)

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.]
T. H. Willcocks, Problem 7795 & solution, Fairy Chess Review 7 (1948) 97, 106.

S. E. Anderson, Compound Perfect Squared Squares (complete to order 36).
S. E. Anderson, Compound Perfect Squared Squares of the Order Twenties, 2013; arXiv:1303.0599 [math.CO], 2013.
Stuart E Anderson, CPSS discoveries attributed to the person or people who found them
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, 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.]
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.]
Lorenz Milla, Depiction of all CPSS until order 31
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Wikipedia, Squaring the square
Index entries for squared squares

Cf. A217155 (counts symmetries of subrectangles as distinct).

Cf. A006983, A181735, A217152, A217153.

Corrected last term from 142 to 143 to include cpss 1170C, added cross reference
Corrected last term from 143 to 144 to include cpss 1224d, incorrectly excluded as a duplicate in the initial count.
Corrected last term from 144 back to 143 after a recount from the original graphs established a bijection between exactly 948 non-isomorphic graphs and 948 isomers in 143 different CPSS arrangements. Gave usual bouwkampcode notation in examples. Removed redundant word "mathematically" from comments. - Stuart E Anderson, Jan 2012
Clarified the definition of 'number' in relation to the 'number' of compound squares, included the definition of 'perfect'. Excluded the trivial dissection from the sequence count. - Stuart E Anderson, May 2012
Definition corrected and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012
a(29) added by Stuart E Anderson, Nov 30 2012
a(30) added by Stuart E Anderson, May 26 2013
a(31)-a(32) added by Stuart E Anderson, Sep 29 2013
a(33)-a(36), enumeration of these orders was completed by Jim Williams in 2014, added by Stuart E Anderson, May 02 2016
a(37)-a(39), enumeration of these orders was completed by Jim Williams in 2018, added by Stuart E Anderson, Sep 17 2018

A217149 Largest 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, 192, 332, 479, 661, 825, 1179, 1544, 2134, 2710, 3641, 4988, 6391, 8430, 11216, 15039, 20242

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. The order of a squared rectangle is the number of constituent squares. By convention the sides of the subsquares are integers with no common factor.

A squared rectangle is simple if it does not contain a smaller squared rectangle. Every perfect square with the largest known side length 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
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection

Cf. A006983, A089047, A129947, A181735, A217148, A217156.

a(29) from Stuart E Anderson added by Geoffrey H. Morley, Nov 23 2012
a(30), a(31), a(32) from Lorenz Milla and Stuart E Anderson, added by 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

A217154 Number of perfect squared rectangles of order n up to symmetries of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 14, 62, 235, 821, 2868, 10193, 36404, 130174, 466913, 1681999, 6083873

Offset: 1

Geoffrey H. Morley, Sep 27 2012

A squared rectangle is simple if it does not contain a smaller squared rectangle, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

			a(10) = 14 comprises the A002839(10) = 6 simple perfect squared rectangles (SPSRs) of order 10 and the 8 trivially compound perfect squared rectangles which each comprises one of the two order 9 SPSRs and one other square.

See crossrefs for references and links.

Cf. A110148 (counts symmetries of any squared subrectangles as equivalent).

Cf. A181735, A217156.

a(n) = A002839(n) + A217153(n) + A217375(n).

a(n) >= 2*a(n-1) + A002839(n) + 2*A002839(n-1) + A217153(n) + 2*A217153(n-1), with equality for n<19.

a(19) and a(20) corrected by Geoffrey H. Morley, Oct 12 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

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

Original entry on oeis.org

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A217156 Number of perfect squared squares of order n up to symmetries of the square.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A129947 Smallest possible side length for a simple perfect squared square of order n; or 0 if no such square exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A002962 Number of simple imperfect squared squares of order n up to symmetry.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A181340 Number of compound perfect squared squares of order n up to symmetries of the square and its squared subrectangles.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A217149 Largest possible side length for a perfect squared square of order n; or 0 if no such square exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A217154 Number of perfect squared rectangles of order n up to symmetries of the rectangle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

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

Views

Author