%I #52 Nov 11 2023 10:39:26
%S 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,
%T 10209,26234,71892,196357,528866,1420439,3784262,10012056,26048712
%N Number of perfect squared squares of order n up to symmetries of the square.
%C 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.
%D H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991, section C2, pp. 81-83.
%D A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
%D 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.
%D 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.
%D J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
%D I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.
%D 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.
%D W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 1-11, Clarendon Press, Oxford, 1998.
%H S. E. Anderson, <a href="http://www.squaring.net/sq/ss/spss/spss.html">Simple Perfect Squared Squares (complete to order 29)</a>.
%H S. E. Anderson, <a href="http://www.squaring.net/sq/ss/cpss/cpss.html">Compound Perfect Squared Squares (complete to order 29)</a>.
%H J. A. Bondy and U. S. R. Murty, <a href="http://book.huihoo.com/pdf/graph-theory-With-applications/pdf/chapter12.pdf">Chapter 12: The Cycle Space and Bond Space</a>, pp. 212-226 in: Graph theory with applications, Elsevier Science Ltd/North-Holland, 1976.
%H C. J. Bouwkamp, <a href="http://dx.doi.org/10.1016/0012-365X(92)90531-J">On some new simple perfect squared squares</a>, Discrete Math. 106-107 (1992), 67-75.
%H C. J. Bouwkamp and A. J. W. Duijvestijn, <a href="http://alexandria.tue.nl/repository/books/391207.pdf">Catalogue of Simple Perfect Squared Squares of orders 21 through 25</a>, EUT Report 92-WSK-03, Eindhoven University of Technology, Eindhoven, The Netherlands, November 1992.
%H C. J. Bouwkamp and A. J. W. Duijvestijn, <a href="http://alexandria.tue.nl/repository/books/430534.pdf">Album of Simple Perfect Squared Squares of order 26</a>, EUT Report 94-WSK-02, Eindhoven University of Technology, Eindhoven, The Netherlands, December 1994.
%H G. Brinkmann and B. D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/papers/plantri-full.pdf">Fast generation of planar graphs</a>, MATCH Commun. Math. Comput. Chem., 58 (2007), 323-357.
%H Gunnar Brinkmann and Brendan McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">plantri and fullgen</a> programs for generation of certain types of planar graph.
%H Gunnar Brinkmann and Brendan McKay, <a href="/A000103/a000103_1.pdf">plantri and fullgen</a> programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
%H R. L. Brooks, C. A. B. Smith, A. H. Stone, and W. T. Tutte, <a href="http://ebookbrowse.com/brooks-smith-stone-tutte-the-dissection-of-rectangles-into-squares-1940-pdf-d37213504">The dissection of rectangles into squares</a>, 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.
%H A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in <a href="http://alexandria.tue.nl/repository/books/44157.pdf ">Philips Res. Rep. 17 (1962), 523-612</a>.
%H A. J. W. Duijvestijn, P. J. Federico, and P. Leeuw, <a href="http://www.jstor.org/stable/2320990">Compound perfect squares</a>, Amer. Math. Monthly 89 (1982), 15-32. [The lowest order of a compound perfect square is 24.]
%H A. J. W. Duijvestijn, <a href="http://doc.utwente.nl/17948/1/Duijvestijn93simple.pdf">Simple perfect squared squares and 2x1 squared rectangles of orders 21 to 24</a>, J. Combin. Theory Ser. B 59 (1993), 26-34.
%H A. J. W. Duijvestijn, <a href="http://dx.doi.org/10.1090/S0025-5718-1994-1208220-9">Simple perfect squared squares and 2x1 squared rectangles of order 25</a>, Math. Comp. 62 (1994), 325-332.
%H A. J. W. Duijvestijn, <a href="http://dx.doi.org/10.1090/S0025-5718-96-00705-3">Simple perfect squares and 2x1 squared rectangles of order 26</a>, Math. Comp. 65 (1996), 1359-1364. [<a href="http://www.squaring.net/downloads/TableI">TableI List of Simple Perfect Squared Squares of order 26</a> and <a href="http://www.squaring.net/downloads/TableII">TableII List of Simple Perfect Squared 2x1 Rectangles of order 26</a> are now on squaring.net and no longer located as described in the paper.]
%H I. Gambini, <a href="http://alain.colmerauer.free.fr/alcol/ArchivesPublications/Gambini/carres.pdf">Quant aux carrés carrelés</a>, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 25.
%H C. A. B. Smith and W. T. Tutte, <a href="https://doi.org/10.4153/CJM-1950-017-8">A class of self-dual maps</a>, Can. J. Math., 2 (1950), 179-196.
%H W. T. Tutte, <a href="https://doi.org/10.4153/CJM-1950-018-5">Squaring the square</a>, Can. J. Math., 2 (1950), 197-209.
%H W. T. Tutte, <a href="https://www.jstor.org/stable/2313308">The quest of the perfect square</a>, Amer. Math. Monthly 72 (1965), 29-35.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectSquareDissection.html">Perfect Square Dissection</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Squaring_the_square">Squaring the square</a>
%H <a href="/index/Sq#squared_squares">Index entries for squared squares</a>
%F a(n) = A006983(n) + A217155(n).
%e a(21) = 1 because there is a unique perfect squared square of order 21. A014530 gives the sizes of its constituent squares.
%Y Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).
%Y Cf. A110148, A217154.
%K nonn,hard,nice,more
%O 1,22
%A _Geoffrey H. Morley_, Sep 27 2012
%E Added a(29) = 10209, _Stuart E Anderson_, Nov 30 2012
%E Added a(30) = 26234, _Stuart E Anderson_, May 26 2013
%E Added a(31) = 71892, a(32) = 196357, _Stuart E Anderson_, Sep 30 2013
%E 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
%E a(36) and a(37) completed by Jim Williams in 2016 to 2018, added by _Stuart E Anderson_, Oct 28 2020