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%I M1658 N0650 #121 Feb 05 2024 11:00:12
%S 0,0,0,0,0,0,0,0,2,6,22,67,213,744,2609,9016,31426,110381,390223,
%T 1383905,4931308,17633773,63301427,228130926
%N Number of simple perfect squared rectangles of order n up to symmetry.
%C A squared rectangle is a rectangle dissected into a finite number of integer-sized squares. If no two of these squares are the same size then the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle or squared square. The order of a squared rectangle is the number of squares into which it is dissected. [Edited by _Stuart E Anderson_, Feb 03 2024]
%D See A217156 for further references and links.
%D C. J. Bouwkamp, personal communication.
%D C. J. Bouwkamp, A. J. W. Duijvestijn and P. Medema, Catalogue of simple squared rectangles of orders nine through fourteen and their elements, Technische Hogeschool, Eindhoven, The Netherlands, May 1960, 50 pp.
%D C. J. Bouwkamp, A. J. W. Duijvestijn and J. Haubrich, Catalogue of simple perfect squared rectangles of orders 9 through 18, Philips Research Laboratories, Eindhoven, The Netherlands, 1964 (unpublished) vols 1-12, 3090 pp.
%D A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
%D M. E. Lines, Think of a Number, Institute of Physics, London, 1990, p. 43.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%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 [sequence p. 207], and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-7 [sequence p. 162].
%H S. E. Anderson, <a href="http://www.squaring.net">Perfect Squared Rectangles and Squared Squares</a>.
%H C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, <a href="http://www.dwc.knaw.nl/DL/publications/PU00018283.pdf">Paper I</a>, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); <a href="http://www.dwc.knaw.nl/DL/publications/PU00018294.pdf">Paper II</a>, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); <a href="http://www.dwc.knaw.nl/DL/publications/PU00018295.pdf">Paper III</a>, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63).
%H C. J. Bouwkamp, <a href="http://www.dwc.knaw.nl/DL/publications/PU00018444.pdf">On the construction of simple perfect squared squares</a>, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625).
%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, A. J. W. Duijvestijn and P. Medema, Tables relating to simple squared rectangles of orders nine through fifteen, Technische Hogeschool, Eindhoven, The Netherlands, August 1960, ii + 360 pp. Reprinted in <a href="http://alexandria.tue.nl/repository/books/150593.pdf">EUT Report 86-WSK-03, January 1986</a>. [Sequence p. i.]
%H C. J. Bouwkamp & N. J. A. Sloane, <a href="/A000162/a000162.pdf">Correspondence, 1971</a>.
%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. [Pp. 324-5 of the original article have counts up to a(12).]
%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 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. 24. [Number of simple rectangles excludes squares in separate column (from order 21).]
%H D. Moews, <a href="http://djm.cc/squared-rectangles.html">Squared rectangles</a>
%H W. T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1963-029-x">A Census of Planar Maps</a>, Canad. J. Math. 15 (1963), 249-271.
%H J. H. van Lint, <a href="/A002839/a002839_1.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectSquareDissection.html">Perfect Square Dissection</a>.
%H <a href="/index/Sq#squared_rectangles">Index entries for squared rectangles</a>
%H <a href="/index/Sq#squared_squares">Index entries for squared squares</a>
%F From _Stuart E Anderson_, Mar 02 2011, Feb 03 2024: (Start)
%F In "A Census of Planar Maps", p. 267, William Tutte gave a conjectured asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order):
%F Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). (End)
%F a(n) = A006983(n) + A219766(n). - _Stuart E Anderson_, Dec 07 2012
%Y Cf. A006983, A002962, A002881, A181735.
%Y Cf. A217153, A217154, A217156, A219766.
%K nonn,nice,hard,more
%O 1,9
%A _N. J. A. Sloane_
%E Definition corrected to include 'simple'. 'Simple' and 'perfect' defined in comments. - _Geoffrey H. Morley_, Mar 11 2010
%E Corrected a(18) and extended terms to order 21. All 3-connected planar graphs up to 22 edges used to generate dissections. Imperfect squared rectangles, compound squared rectangles, and all squared squares filtered out leaving simple perfect squared rectangles. - _Stuart E Anderson_, Mar 2011
%E Corrected a(18) to a(21) after removing last remaining compounds. - _Stuart E Anderson_, Apr 10 2011
%E Added a(22), a(23) and a(24) from Ian Gambini's thesis and corrected a(22). Added I. Gambini's thesis reference. - _Stuart E Anderson_, May 08 2011
%E Added some additional references, previous correction to a(22) is an increase of 4 based on a new count of order 22. - _Stuart E Anderson_, Jul 13 2012
%E Terms a(21)-a(24) corrected to include squares by _Geoffrey H. Morley_, Oct 17 2012
%E a(22)=17633773 from _Stuart E Anderson_ confirmed by _Geoffrey H. Morley_, Nov 28 2012
%E a(23)-a(24) from Gambini confirmed by _Stuart E Anderson_, Dec 07 2012
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