A129947 -id:A129947 - 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.

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

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

A349205 a(n) is the side length (size) of the smallest element in a simple perfect squared square of order n such that the ratio of the size of the smallest element to the total size of the square assumes a maximum over all possible A006983(n) dissections of order n.

Original entry on oeis.org

2, 4, 2, 3, 12, 17, 48, 29, 62, 53, 64, 156, 70, 270, 257, 333, 716

Offset: 21

Hugo Pfoertner, Nov 22 2021

			See Pfoertner link.

Stuart E. Anderson, Catalogues of Perfect Squared Squares.
Hugo Pfoertner, Data of squares with maximum ratio, (Nov 2021).
Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

A349206 gives the corresponding total sizes of those squares that lead to the maximum ratio.

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

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.

A349207 a(n) is the side length (size) of the smallest element in a simple perfect squared square of order n such that the ratio of the size of the smallest element to the size of the largest element of the square assumes a maximum over all possible A006983(n) dissections of order n.

Original entry on oeis.org

2, 4, 2, 3, 16, 17, 48, 29, 62, 69, 64, 88, 70, 111, 355, 333, 543

Offset: 21

Hugo Pfoertner, Nov 22 2021

			See Pfoertner link.

Stuart E. Anderson, Catalogues of Perfect Squared Squares.
Hugo Pfoertner, Data of squares with maximum ratio, (Nov 2021).
Rainer Rosenthal, Illustration of sequence terms, Nov 2021.

A349208 gives the corresponding sizes of the largest elements that lead to the maximum ratio.

Cf. A006983, A129947, A217149, A228953, A349205, A349206, 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.

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

cp's OEIS Frontend

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

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

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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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A349205 a(n) is the side length (size) of the smallest element in a simple perfect squared square of order n such that the ratio of the size of the smallest element to the total size of the square assumes a maximum over all possible A006983(n) dissections of order n.

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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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A349207 a(n) is the side length (size) of the smallest element in a simple perfect squared square of order n such that the ratio of the size of the smallest element to the size of the largest element of the square assumes a maximum over all possible A006983(n) dissections of order n.

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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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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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