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A217155 Number of compound perfect squared squares of order n up to symmetries of the square. 6
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 (list; graph; refs; listen; history; text; internal format)



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


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.]


Table of n, a(n) for n=1..39.

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


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.


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

Cf. A006983, A217152, A217153, A217156.

Sequence in context: A287596 A268363 A038053 * A120267 A012278 A070040

Adjacent sequences:  A217152 A217153 A217154 * A217156 A217157 A217158




Geoffrey H. Morley, Sep 27 2012


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



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Last modified November 18 11:26 EST 2018. Contains 317302 sequences. (Running on oeis4.)