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%I #68 Oct 27 2020 08:44:02
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,112,110,110,120,147,212,180,
%T 201,221,201,215,185,223,218,225,253,237
%N Smallest possible side length for a simple perfect squared square of order n; or 0 if no such square exists.
%C It is not known whether this sequence is the same as sequence A217148. It may be that A129947(33) = 246 and A217148(33) = 234. - _Geoffrey H. Morley_, Jan 10 2013
%C From _Geoffrey H. Morley_, Oct 17 2012: (Start)
%C 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. The order of a squared rectangle is the number of constituent squares.
%C A squared rectangle is simple if it does not contain a smaller squared rectangle.
%C The smallest known sides of simple perfect squared squares (and the known orders of the squares) are 110 (22, 23), 112 (21), 120 (24), 139 (22, 23), 140 (23), 145 (23), 147 (22, 25) ...
%C The upper bounds shown below for n = 38 and 40-44 are from J. B. Williams. The rest are from Gambini's thesis. - _Geoffrey H. Morley_, Mar 08 2013
%C ======================================
%C Upper bounds for a(n) for n = 38 to 59
%C ======================================
%C | +0 +1 +2 +3 +4 +5 +6 +7 +8 +9
%C ======================================================
%C 30 | - - - - - - - - 352 360
%C 40 | 328 336 360 413 425 543 601 691 621 779
%C 50 | 788 853 ? 824 971 939 929 985 1100 1060
%C ======================================================
%C (End)
%H S. E. Anderson, <a href="http://www.squaring.net/">Perfect Squared Rectangles and Squared Squares</a>.
%H Stuart E. Anderson, <a href="http://www.squaring.net/sq/ss/s-pss.html">'Special' Perfect Squared Squares"</a>, accessed 2014. - _N. J. A. Sloane_, Mar 30 2014
%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, pp. 73-78.
%H Ed Pegg Jr., <a href="https://community.wolfram.com/groups/-/m/t/2044450">Advances in Squared Squares</a>, Wolfram Community Bulletin, Jul 23 2020
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectSquareDissection.html">Perfect Square Dissection</a>
%H <a href="/index/Sq#squared_squares">Index entries for squared squares</a>
%Y Cf. A006983, A174386, A181735, A217148, A217149, A217156.
%K nonn,hard,more
%O 1,21
%A _Alexander Adamchuk_, Jun 09 2007, corrected Jun 11 2007
%E Unproved statement misattributed to Skinner replaced, known upper bounds corrected, and crossref added by _Geoffrey H. Morley_, Mar 19 2010
%E Additional term added, initial term a(0)=1 deleted by _Stuart E Anderson_, Dec 26 2010
%E Upper bounds for terms a(31) to a(78), (all from Ian Gambini's thesis) added by _Stuart E Anderson_, Jan 20 2011
%E New bound for a(29)<=221, from _Stuart E Anderson_ & _Ed Pegg Jr_, Jan 20 2011
%E a(29) confirmed as 221, from _Stuart E Anderson_, _Ed Pegg Jr_, and Stephen Johnson, Aug 22 2011
%E New bound for a(31)<=236, computed by Stephen Johnson in September 2011, updated by _Stuart E Anderson_, Oct 04 2011
%E a(30) from _Stuart E Anderson_ and Lorenz Milla added by _Geoffrey H. Morley_, Jun 15 2013
%E a(31) and a(32) from Lorenz Milla and _Stuart E Anderson_, Oct 05 2013
%E For additional terms see the Ed Pegg link, also A006983. - _N. J. A. Sloane_, Jul 29 2020
%E a(33) to a(37) from J. B. Williams, added by _Stuart E Anderson_, Oct 27 2020
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