A181735 - OEIS

%I #64 Oct 27 2020 12:49:18

%S 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,

%T 8313,21507,57329,152102,400610,1053254,2750411,7140575,18326660

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

%C 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

%D See A217156 for further references and links.

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

%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 28)</a>.

%H C. J. Bouwkamp, On some new simple perfect squared squares, Discrete Math. 106-107 (1992), 67-75. <a href="http://dx.doi.org/10.1016/0012-365X(92)90531-J">doi:10.1016/0012-365X(92)90531-J</a>

%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 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, Simple perfect squared squares and 2x1 squared rectangles of order 25, Math. Comp. 62 (1994), 325-332. <a href="http://dx.doi.org/10.1090/S0025-5718-1994-1208220-9">doi:10.1090/S0025-5718-1994-1208220-9</a>

%H A. J. W. Duijvestijn, Simple perfect squares and 2x1 squared rectangles of order 26, Math. Comp. 65 (1996), 1359-1364. <a href="http://dx.doi.org/10.1090/S0025-5718-96-00705-3">doi:10.1090/S0025-5718-96-00705-3</a> [<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. [But symmetries of any subrectangles counted as distinct.]

%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) + A181340(n). - _Geoffrey H. Morley_, Oct 17 2012

%e From _Geoffrey H. Morley_, Oct 17 2012 (Start):

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

%e 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)

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

%Y Cf. A110148, A217154.

%K nonn,more,hard

%O 1,22

%A _Stuart E Anderson_

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

%E Added more clarification in comments on definition of a perfect squared square. - _Stuart E Anderson_, May 23 2012

%E Definition corrected and offset changed to 1 by _Geoffrey H. Morley_, Oct 17 2012

%E a(29) added by _Stuart E Anderson_, Dec 01 2012

%E a(30) added by _Stuart E Anderson_, May 26 2013

%E a(31) and a(32) added by _Stuart E Anderson_, Sep 30 2013

%E a(33), a(34) and a(35) added after enumeration by Jim Williams, _Stuart E Anderson_, May 02 2016

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