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A002839
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Number of simple perfect squared rectangles of order n.
(Formerly M1658 N0650)
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9
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0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633769, 63301415, 228130900
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,9
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COMMENTS
| The order of a squared rectangle is the number of squares into which it is divided. It is simple if it contains no smaller squared rectangle. It is perfect if no two squares are the same size.
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REFERENCES
| C. J. Bouwkamp, personal communication.
C. J. Bouwkamp and A. J. W. Duijvestijn, Catalogue of Simple Perfect Squared Squares of Orders 21 Through 25. Eindhoven Univ. Technology, Dept. of Math., Report 92-WSK-03, Nov. 1992.
C. J. Bouwkamp and A. J. W. Duijvestijn, Album of Simple Perfect Squared Squares of order 26, Eindhoven University of Technology, Faculty of Mathematics and Computing Science, EUT Report 94-WSK-02, December 1994.
M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions. Simon and Schuster, NY, 1961, p. 207.
M. E. Lines, Think of a Number, pp. 43 Institute of Physics, London 1990.
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).
Gambini, Ian. Thesis; 'Quant aux carrés carrelés' L’Universite de la Mediterranee Aix-Marseille II 1999.
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LINKS
| S. E. Anderson, Perfect Squared Rectangles and Squared Squares
D. Moews, Squared rectangles
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| William Tutte gave an asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order);
a(n) = ((n^(-5/2))*(4^n))/(2^5*sqrt(pi))
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CROSSREFS
| Cf. A006983, A002962, A002881, A014530, A181735
Sequence in context: A147800 A027561 A126171 * A109194 A014334 A107239
Adjacent sequences: A002836 A002837 A002838 * A002840 A002841 A002842
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KEYWORD
| nonn,nice,hard
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Definition corrected to include 'simple'. 'Simple' and 'perfect' defined in comments Geoffrey H. Morley (ghmorley(AT)googlemail.com), 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 stuart.errol.anderson (AT) gmail.com, Mar 2011. Corrected a(18) to a(21) after removing last remaining compounds. Stuart E Anderson Apr 2011. Added a(22), a(23) and a(24) from Ian Gambini's thesis and corrected a(22), reducing the count by 4. Added I. Gambini's thesis reference.
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