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A219766 Number of nonsquare simple perfect squared rectangles of order n up to symmetry. 3
0, 0, 0, 0, 0, 0, 0, 0, 2, 6, 22, 67, 213, 744, 2609, 9016, 31426, 110381, 390223, 1383905, 4931307, 17633765, 63301415, 228130900 (list; graph; refs; listen; history; text; internal format)



A squared rectangle (which may be a square, but not in this particular sequence) 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.


See A006983 and A217156 for references.


Table of n, a(n) for n=1..24.

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.

See A006983 and A217156 for further links.


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

In 'A Census of Planar Maps', William Tutte gave an asymptotic formula for the number of perfect squared rectangles where n is the number of elements in the dissection (the order):

a(n) ~ ((n^(-5/2))*(4^n))/(2^5*sqrt(Pi)).


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 *)


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

Cf. A217153, A217154, A217156.

Sequence in context: A321626 A126171 A228396 * A002839 A109194 A014334

Adjacent sequences:  A219763 A219764 A219765 * A219767 A219768 A219769




Stuart E Anderson, Nov 27 2012


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



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Last modified September 18 20:19 EDT 2020. Contains 337173 sequences. (Running on oeis4.)