

A181735


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


13



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, 8313, 21507, 57329, 152102, 400610, 1053254, 2750411, 7140575, 18326660
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OFFSET

1,22


COMMENTS

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


REFERENCES

See A217156 for further references and links.
J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993.


LINKS

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]
A. J. W. Duijvestijn, P. J. Federico and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 1532. [The lowest order of a compound perfect square is 24.]
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée AixMarseille II, 1999, p. 25. [But symmetries of any subrectangles counted as distinct.]


FORMULA



EXAMPLE

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


CROSSREFS

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


KEYWORD

nonn,more,hard


AUTHOR



EXTENSIONS

Corrected last term to 3144 to reflect correction to 143 of last order 28 compound squares term in A181340.
Added more clarification in comments on definition of a perfect squared square.  Stuart E Anderson, May 23 2012
a(33), a(34) and a(35) added after enumeration by Jim Williams, Stuart E Anderson, May 02 2016
a(36) and a(37) from Jim Williams, completed in 2018 to 2020, added by Stuart E Anderson, Oct 28 2020


STATUS

approved



