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A002839 Number of simple perfect squared rectangles of order n up to symmetry.
(Formerly M1658 N0650)
18
0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931308, 17633773, 63301427, 228130926, 825229110 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
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]
REFERENCES
See A217156 for further references and links.
C. J. Bouwkamp, personal communication.
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.
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perfect squared rectangles of orders 9 through 18, Philips Research Laboratories, Eindhoven, The Netherlands, 1964 (unpublished) vols 1-12, 3090 pp.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
M. E. Lines, Think of a Number, Institute of Physics, London, 1990, p. 43.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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].
LINKS
C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63).
C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625).
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, 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 EUT Report 86-WSK-03, January 1986. [Sequence p. i.]
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971.
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. [Pp. 324-5 of the original article have counts up to a(12).]
A. J. W. Duijvestijn, Electronic Computation of Squared Rectangles, Thesis, Technische Hogeschool, Eindhoven, Netherlands, 1962. Reprinted in Philips Res. Rep. 17 (1962), 523-612.
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24. [Number of simple rectangles excludes squares in separate column (from order 21).]
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
Eric Weisstein's World of Mathematics, Perfect Square Dissection.
FORMULA
From Stuart E Anderson, Mar 02 2011, Feb 03 2024: (Start)
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):
Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). (End)
a(n) = A006983(n) + A219766(n). - Stuart E Anderson, Dec 07 2012
CROSSREFS
Sequence in context: A126171 A228396 A219766 * A109194 A014334 A107239
KEYWORD
nonn,nice,hard,more
AUTHOR
EXTENSIONS
Definition corrected to include 'simple'. 'Simple' and 'perfect' defined in comments. - Geoffrey H. Morley, Mar 11 2010
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
Corrected a(18) to a(21) after removing last remaining compounds. - Stuart E Anderson, Apr 10 2011
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
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
Terms a(21)-a(24) corrected to include squares by Geoffrey H. Morley, Oct 17 2012
a(22)=17633773 from Stuart E Anderson confirmed by Geoffrey H. Morley, Nov 28 2012
a(23)-a(24) from Gambini confirmed by Stuart E Anderson, Dec 07 2012
a(25) from Stuart E Anderson, May 07 2024
STATUS
approved

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Last modified July 25 11:18 EDT 2024. Contains 374588 sequences. (Running on oeis4.)