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%I #70 Feb 04 2024 15:50:10
%S 0,0,0,0,0,0,0,0,2,6,22,67,213,744,2609,9016,31426,110381,390223,
%T 1383905,4931307,17633765,63301415,228130900
%N Number of nonsquare simple perfect 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 02 2024]
%D See A006983 and A217156 for references.
%H Stuart E Anderson, <a href="http://www.squaring.net/sq/sr/spsr/spsr.html">Simple Perfect Squared Rectangles</a>. [Nonsquare rectangles only]
%H I. Gambini, <a href="http://alain.colmerauer.free.fr/alcol/ArchivesPublications/Gambini/carres.pdf">Quant aux carrés carrelés</a>, Thesis, Université de la Méditerranée Aix-Marseille II, 1999, p. 24.
%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.
%H See A006983 and A217156 for further links.
%F a(n) = A002839(n) - A006983(n).
%F In "A Census of Planar Maps", p. 267, William Tutte gave a conjectured asymptotic formula for the number, a(n) of perfect squared rectangles of order n:
%F Conjectured: a(n) ~ n^(-5/2) * 4^n / (243*sqrt(Pi)). [Corrected by _Stuart E Anderson_, Feb 02 2024]
%t A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
%t A002839 = A@002839;
%t A006983 = A@006983;
%t a[n_] := A002839[[n]] - A006983[[n]];
%t a /@ Range[24] (* _Jean-François Alcover_, Jan 13 2020 *)
%Y Cf. A002839, A006983, A002962, A002881, A181735.
%Y Cf. A217153, A217154, A217156.
%K nonn,hard,more
%O 1,9
%A _Stuart E Anderson_, Nov 27 2012
%E a(9)-a(24) enumerated by Gambini 1999, confirmed by _Stuart E Anderson_, Dec 07 2012
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