A181735 Number of perfect squared squares of order n up to symmetries of the square and of its squared subrectangles, if any.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 27, 162, 457, 1198, 3144, 8313, 21507, 57329, 152102, 400610, 1053254, 2750411, 7140575, 18326660

Offset: 1

Author

Stuart E Anderson

Keywords

• nonn
• more
• hard

Comments

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

Examples

			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)
		

References

• See A217156 for further references and links.
• J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.

Links

• 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.
• 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.]
• 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. [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

Crossrefs

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

Cf. A110148, A217154.

Formula

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

Extensions

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

a(33), a(34) and a(35) added after enumeration by Jim Williams, Stuart E Anderson, May 02 2016

a(36) and a(37) from Jim Williams, completed in 2018 to 2020, added by Stuart E Anderson, Oct 28 2020