A349208 -id:A349208 - 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.

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.

cp's OEIS Frontend

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

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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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