A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry.

0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633765, 63301415, 228130900, 825228950, 2994833413

Offset: 1

Author

Stuart E Anderson, Nov 27 2012

Keywords

• nonn
• hard
• more

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 02 2024]

References

• See A006983 and A217156 for references.

Links

• Stuart E Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
• I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24.
• W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.
• See A006983 and A217156 for further links.

Crossrefs

Cf. A002839, A006983, A002962, A002881, A181735.

Cf. A217153, A217154, A217156.

Programs

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A002839 = A@002839;
A006983 = A@006983;
a[n_] := A002839[[n]] - A006983[[n]];
a /@ Range[24] (* Jean-François Alcover, Jan 13 2020 *)

Formula

a(n) = A002839(n) - A006983(n).

In "A Census of Planar Maps", p. 267, William Tutte gave a conjectured asymptotic formula for the number, a(n) of perfect squared rectangles of order n:

Conjectured: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). [Corrected by Stuart E Anderson, Feb 02 2024]

Extensions

a(9)-a(24) enumerated by Gambini 1999, confirmed by Stuart E Anderson, Dec 07 2012

a(25) from Stuart E Anderson, May 07 2024

a(26) from Stuart E Anderson, Jul 28 2024