A220167 Number of simple squared rectangles of order n up to symmetry.

3, 6, 22, 76, 247, 848, 2892, 9969, 34455, 119894, 420582, 1482874, 5254954, 18714432, 66969859, 240739417

Offset: 1

Author

Stuart E Anderson, Dec 06 2012

Keywords

• nonn
• hard

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 03 2024]

References

• See A006983 and A217156 for references and links.

Links

• S. E. Anderson, Simple Perfect Squared Rectangles. [Nonsquare rectangles only]
• S. E. Anderson, Simple Perfect Squared Squares.
• S. E. Anderson, Simple Imperfect Squared Rectangles. [Nonsquare rectangles only]
• S. E. Anderson, Simple Imperfect Squared Squares.
• W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Crossrefs

Cf. A002839, A002881, A006983, A002962, A220165, A219766.

Formula

a(n) = A002839(n) + A002881(n).

a(n) = A006983(n) + A002962(n) + A220165(n) + A219766(n).

Conjecture: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)), from "A Census of Planar Maps", p. 267, where William Tutte gave a conjectured asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order). [Corrected by Stuart E Anderson, Feb 03 2024]

Extensions

a(9)-a(24) from Stuart E Anderson, Dec 07 2012