A217156 Number of perfect squared squares of order n up to symmetries of the square.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 12, 30, 172, 541, 1372, 3949, 10209, 26234, 71892, 196357, 528866, 1420439, 3784262, 10012056, 26048712

Offset: 1

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

• nonn
• hard
• nice
• more

Comments

a(n) is the number of solutions to the classic problem of 'squaring the square' by n unequal squares. 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.

Examples

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

References

• H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991, section C2, pp. 81-83.
• A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
• P. J. Federico, Squaring rectangles and squares: A historical review with annotated bibliography, in Graph Theory and Related Topics, J. A. Bondy and U. S. R. Murty, eds., Academic Press, 1979, 173-196.
• J. H. van Lint and R. M. Wilson, A course in combinatorics, Chapter 34 "Electrical networks and squared squares", pp. 449-460, Cambridge Univ. Press, 1992.
• J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
• I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.
• W. T. Tutte, Squaring the Square, in M. Gardner's 'Mathematical Games' column in Scientific American 199, Nov. 1958, pp. 136-142, 166. Reprinted with addendum and bibliography in the US in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, Simon and Schuster, New York (1961), pp. 186-209, 250, and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-7.
• W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 1-11, Clarendon Press, Oxford, 1998.

Links

• S. E. Anderson, Simple Perfect Squared Squares (complete to order 29).
• S. E. Anderson, Compound Perfect Squared Squares (complete to order 29).
• J. A. Bondy and U. S. R. Murty, Chapter 12: The Cycle Space and Bond Space, pp. 212-226 in: Graph theory with applications, Elsevier Science Ltd/North-Holland, 1976.
• 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.
• 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]
• R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, The dissection of rectangles into squares, Duke Math. J., 7 (1940), 312-340. Reprinted in I. Gessel and G.-C. Rota (editors), Classic papers in combinatorics, Birkhäuser Boston, 1987, pp. 88-116.
• A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523-612.
• 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.
• A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. [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.
• C. A. B. Smith and W. T. Tutte, A class of self-dual maps, Can. J. Math., 2 (1950), 179-196.
• W. T. Tutte, Squaring the square, Can. J. Math., 2 (1950), 197-209.
• W. T. Tutte, The quest of the perfect square, Amer. Math. Monthly 72 (1965), 29-35.
• Eric Weisstein's World of Mathematics, Perfect Square Dissection
• Wikipedia, Squaring the square
• Index entries for squared squares

Crossrefs

Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).

Cf. A110148, A217154.

Formula

a(n) = A006983(n) + A217155(n).

Extensions

Added a(29) = 10209, Stuart E Anderson, Nov 30 2012

Added a(30) = 26234, Stuart E Anderson, May 26 2013

Added a(31) = 71892, a(32) = 196357, Stuart E Anderson, Sep 30 2013

Added a(33) = 528866, a(34) = 1420439, a(35) = 3784262, due to enumeration completed by Jim Williams in 2014 and 2016. Stuart E Anderson, May 02 2016

a(36) and a(37) completed by Jim Williams in 2016 to 2018, added by Stuart E Anderson, Oct 28 2020