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

Author

N. J. A. Sloane

Keywords

• hard
• nonn

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. 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]

References

• 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].

Links

• 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

Crossrefs

Cf. A006983, A002962, A002839, A220165.

Cf. A181735, A217153, A217154, A217156.

Formula

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

Extensions

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