1,22

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. - Geoffrey H. Morley, Oct 17 2012

See A217156 for further references and links.

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.]

J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.

Table of n, a(n) for n=1..32.

S. E. Anderson, Simple Perfect Squared Squares (complete to order 29).

S. E. Anderson, Compound Perfect Squared Squares (complete to order 28).

C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75. doi:10.1016/0012-365X(92)90531-J

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.

G. Brinkmann and B. D. McKay, plantri and fullgen. [Programs for generation of certain types of planar graph.]

A. J. W. Dujivestijn, 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. Dujivestijn, 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 carres carreles, Thesis, Universite de la Mediterranee Aix-Marseille II, 1999, p. 25. [But symmetries of any subrectangles counted as distinct.]

Eric Weisstein's World of Mathematics, Perfect Square Dissection

Wikipedia, Squaring the square

Index entries for squared squares

a(n) = A006983(n) + A181340(n). - Geoffrey H. Morley, Oct 17 2012

From Geoffrey H. Morley, Oct 17 2012 (Start):

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

a(24) = 27 because there are A217156(24) = 30 perfect squared squares of order 24 but four of them differ only in the symmetries of a squared subrectangle. (End)

Cf. A217156 (counts symmetries of any subrectangles as distinct).

Cf. A110148, A217154.

Sequence in context: A006983 A072327 A218558 * A161415 A117802 A083485

Adjacent sequences:  A181732 A181733 A181734 * A181736 A181737 A181738

nonn,more,hard

Stuart E Anderson

Corrected last term to 3144 to reflect correction to 143 of last order 28 compound squares term in A181340.

Added more clarification in comments on definition of a perfect squared square. - Stuart E Anderson, May 23 2012

Definition corrected and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012

a(29) added by Stuart E Anderson, Dec 01 2012

a(30) added by Stuart E Anderson, May 26 2013

a(31) and a(32) added by Stuart E Anderson, Sep 30 2013

approved