A220165 -id:A220165 - cp's OEIS frontend

A002881 Number of simple imperfect squared rectangles of order n up to symmetry.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 9, 34, 104, 283, 953, 3029, 9513, 30359, 98969, 323646, 1080659, 3668432, 12608491, 43745771, 153812801

Offset: 1

N. J. A. Sloane

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. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of its constituent squares. [Geoffrey H. Morley, Oct 17 2012]

C. J. Bouwkamp, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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 [sequence on p. 207], and in the UK in M. Gardner, More Mathematical Puzzles and Diversions, Bell (1963) and Penguin Books (1966), pp. 146-164, 186-187 [sequence on p. 162].

S. E. Anderson, Simple Imperfect Squared Rectangles. [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Squares.
C. J. Bouwkamp, A. J. W. Duijvestijn and P. Medema, Tables relating to simple squared rectangles of orders nine through fifteen, Technische Hogeschool, Eindhoven, The Netherlands, August 1960, ii + 360 pp. Reprinted in EUT Report 86-WSK-03, January 1986. [Sequence p. i.]
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971.
Eric Weisstein's World of Mathematics, Perfect Rectangle.
Index entries for squared rectangles
Index entries for squared squares

Cf. A006983, A002962, A002839, A220165.

Cf. A181735, A217153, A217154, A217156.

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

Edited ("simple" added to the definition, definition of "simple" given in the comments), terms a(13), a(15), a(16), a(17), and a(18) corrected, and terms extended to a(20) by Stuart E Anderson, Mar 09 2011
a(16)-a(20) corrected (excess compounds removed) by Stuart E Anderson, Apr 10 2011
Sequence reverted to the one in Bouwkamp et al. (1960), Gardner (1961), Sloane (1973), and Sloane & Plouffe (1995), which includes simple imperfect squares, by Geoffrey H. Morley, Oct 17 2012
a(19)-a(20) corrected, a(21)-a(24) added by Stuart E Anderson, Dec 03 2012
a(25) from Stuart E Anderson, May 07 2024
a(26) from Stuart E Anderson, Jul 28 2024

A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 3, 6, 22, 76, 246, 848, 2889, 9964, 34440, 119875, 420525, 1482802, 5254679, 18713933, 66968081, 240735712

Offset: 1

Stuart E Anderson, Dec 06 2012

A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of integer sized squares. If no two of these squares have the same size the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. This sequence counts nonsquare simple perfect squared rectangles and nonsquare simple imperfect squared rectangles.

See A006983 and A217156 for further links.

S. E. Anderson, Simple Perfect Squared Rectangles [Nonsquare rectangles only]
S. E. Anderson, Simple Imperfect Squared Rectangles [Nonsquare rectangles only]

Cf. A219766, A220165, A002839, A002881.

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

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

Original entry on oeis.org

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

Offset: 1

Stuart E Anderson, Dec 06 2012

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]

See A006983 and A217156 for references and 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.

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

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]

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

cp's OEIS Frontend

A002881 Number of simple imperfect squared rectangles of order n up to symmetry.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions

A220166 Number of nonsquare simple squared rectangles of order n up to symmetry.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

Extensions