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A217156
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Number of perfect squared squares of order n up to symmetries of the square.
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17
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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, 10209, 26234, 71892, 196357, 528866, 1420439, 3784262, 10012056, 26048712
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OFFSET
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1,22
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COMMENTS
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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.
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REFERENCES
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H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1991, section C2, pp. 81-83.
A. J. W. Duijvestijn, Fast calculation of inverse matrices occurring in squared rectangle calculation, Philips Res. Rep. 30 (1975), 329-339.
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.
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.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.
I. Stewart, Squaring the Square, Scientific Amer., 277, July 1997, pp. 94-96.
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.
W. T. Tutte, Graph theory as I have known it, Chapter 1 "Squaring the square", pp. 1-11, Clarendon Press, Oxford, 1998.
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LINKS
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Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
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.
A. J. W. Duijvestijn, P. J. Federico, and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 15-32. [The lowest order of a compound perfect square is 24.]
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FORMULA
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EXAMPLE
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a(21) = 1 because there is a unique perfect squared square of order 21. A014530 gives the sizes of its constituent squares.
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CROSSREFS
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Cf. A181735 (counts symmetries of any squared subrectangles as equivalent).
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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EXTENSIONS
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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
a(36) and a(37) completed by Jim Williams in 2016 to 2018, added by Stuart E Anderson, Oct 28 2020
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STATUS
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approved
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