A220167 - OEIS

%I #27 Feb 04 2024 15:50:02

%S 3,6,22,76,247,848,2892,9969,34455,119894,420582,1482874,5254954,

%T 18714432,66969859,240739417

%N Number of simple squared rectangles of order n up to symmetry.

%C 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]

%D See A006983 and A217156 for references and links.

%H S. E. Anderson, <a href="http://www.squaring.net/sq/sr/spsr/spsr.html">Simple Perfect Squared Rectangles</a>. [Nonsquare rectangles only]

%H S. E. Anderson, <a href="http://www.squaring.net/sq/ss/spss/spss.html">Simple Perfect Squared Squares</a>.

%H S. E. Anderson, <a href="http://www.squaring.net/sq/sr/sisr/sisr.html">Simple Imperfect Squared Rectangles</a>. [Nonsquare rectangles only]

%H S. E. Anderson, <a href="http://www.squaring.net/sq/ss/siss/siss.html">Simple Imperfect Squared Squares</a>.

%H W. T. Tutte, <a href="http://dx.doi.org/10.4153/CJM-1963-029-x">A Census of Planar Maps</a>, Canad. J. Math. 15 (1963), 249-271.

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

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

%F 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]

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

%K nonn,hard

%O 1,1

%A _Stuart E Anderson_, Dec 06 2012

%E a(9)-a(24) from _Stuart E Anderson_, Dec 07 2012