OFFSET
1,21
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. By convention the sides of the subsquares are integers with no common factor.
A squared rectangle is simple if it does not contain a smaller squared rectangle. Every perfect square with the largest known side length for each order up to 37 is simple.
From Stuart E Anderson, Feb 24 2026: (Start)
Empirical observation: The maximum side length of perfect squared squares scales asymptotically by a factor of approximately 1.36 per order n. This aligns closely with the 1.33 to 1.37 growth factors observed in other classes of squared squares, such as Simple Imperfect Squared Squares (SISS) and Compound Perfect Squared Squares (CPSS).
Notably, squared squares scale at roughly the square root of the rate of simple squared rectangles, which grow by a factor of 1.76 to 1.77 per order. A geometric intuition for this unproven heuristic is that squares are constrained in two dimensions (W = H), limiting their maximum dimension to scale with the square root of the maximum available combinatorial area. In contrast, unconstrained rectangles (W != H) can elongate, allowing their maximum width to scale directly with the area.
The bounds for n > 32 were achieved using an isomorph-free planar graph generator and Cholesky decomposition matrix solver to evaluate the electrical networks (J. B. Williams, 2020).
Empirical observation: Logarithmic regression of the data from n=21 to 37 indicates an asymptotic growth scaling factor of approximately 1.36 per order. A curve fit suggests a(n) approximately equal to exp(0.3054*n - 1.3020). A formal proof of this asymptotic growth rate is not yet known. (End)
LINKS
S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
Stuart Anderson, 'Special' Perfect Squared Squares", accessed 2014. - N. J. A. Sloane, Mar 30 2014
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Geoffrey H. Morley, Sep 27 2012
EXTENSIONS
a(29) from Stuart E Anderson added by Geoffrey H. Morley, Nov 23 2012
a(30), a(31), a(32) from Lorenz Milla and Stuart E Anderson, added by Stuart E Anderson, Oct 05 2013
For additional terms see the Ed Pegg link, also A006983. - N. J. A. Sloane, Jul 29 2020
a(33) to a(37) from J. B. Williams added by Stuart E Anderson, Oct 27 2020
STATUS
approved
