

A217156


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


15



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
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OFFSET

1,22


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.


REFERENCES

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, SpringerVerlag, 1991, section C2, pp. 8183.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329339.
A. J. W. Duijvestijn, P. J. Federico and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 1532. [The lowest order of a compound perfect square is 24.]
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, 173196.
J. H. van Lint and R. M. Wilson, A course in combinatorics, Chapter 34 "Electrical networks and squared squares", pp. 449460, Cambridge Univ. Press, 1992.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
C. A. B. Smith and W. T. Tutte, A class of selfdual maps, Can. J. Math., 2 (1950), 179196.
I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 9496.
W. T. Tutte, Squaring the square, Can. J. Math., 2 (1950), 197209.
W. T. Tutte, Squaring the Square, in M. Gardner's 'Mathematical Games' column in Scientific American 199, Nov. 1958, pp. 136142, 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. 186209, 250, and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146164, 1867.
W. T. Tutte, The quest of the perfect square, Amer. Math. Monthly 72 (1965), 2935.
W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 111, Clarendon Press, Oxford, 1998.


LINKS

Table of n, a(n) for n=1..35.
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. 212226 in: Graph theory with applications, Elsevier Science Ltd/NorthHolland, 1976.
C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106107 (1992), 6775.
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of orders 21 through 25, EUT Report 92WSK03, 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 94WSK02, 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), 323357.
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), 312340. Reprinted in I. Gessel and G.C. Rota (editors), Classic papers in combinatorics, Birkhauser Boston, 1987, pp. 88116.
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523612.
A. J. W. Dujivestijn, Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24, J. Combin. Theory Ser. B 59 (1993), 2634.
A. J. W. Dujivestijn, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325332.
A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 13591364. [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 AixMarseille II, 1999, p. 25.
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Wikipedia, Squaring the square
Index entries for squared squares


FORMULA

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


EXAMPLE

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


CROSSREFS

Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).
Cf. A110148, A217154.
Sequence in context: A256531 A117802 A083485 * A066934 A137148 A211778
Adjacent sequences: A217153 A217154 A217155 * A217157 A217158 A217159


KEYWORD

nonn,hard,nice


AUTHOR

Geoffrey H. Morley, Sep 27 2012


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


STATUS

approved



