

A129947


Smallest possible side length for a simple perfect squared square of order n; or 0 if no such square exists.


12



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, 201, 221, 201, 215, 185, 223, 218, 225, 253, 237
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OFFSET

1,21


COMMENTS

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
From Geoffrey H. Morley, Oct 17 2012: (Start)
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.
A squared rectangle is simple if it does not contain a smaller squared rectangle.
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) ...
The upper bounds shown below for n = 38 and 4044 are from J. B. Williams. The rest are from Gambini's thesis.  Geoffrey H. Morley, Mar 08 2013
======================================
Upper bounds for a(n) for n = 38 to 59
======================================
 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9
======================================================
30          352 360
40  328 336 360 413 425 543 601 691 621 779
50  788 853 ? 824 971 939 929 985 1100 1060
======================================================
(End)


LINKS

Table of n, a(n) for n=1..37.
S. E. Anderson, Perfect Squared Rectangles and Squared Squares.
Stuart E. Anderson, 'Special' Perfect Squared Squares", accessed 2014.  N. J. A. Sloane, Mar 30 2014
I. Gambini, Quant aux carrés carrelés, Thesis, Université de la Méditerranée AixMarseille II, 1999, pp. 7378.
Ed Pegg Jr., Advances in Squared Squares, Wolfram Community Bulletin, Jul 23 2020
Eric Weisstein's World of Mathematics, Perfect Square Dissection
Index entries for squared squares


CROSSREFS

Cf. A006983, A174386, A181735, A217148, A217149, A217156.
Sequence in context: A340470 A010032 A190026 * A217148 A223822 A300006
Adjacent sequences: A129944 A129945 A129946 * A129948 A129949 A129950


KEYWORD

nonn,hard,more


AUTHOR

Alexander Adamchuk, Jun 09 2007, corrected Jun 11 2007


EXTENSIONS

Unproved statement misattributed to Skinner replaced, known upper bounds corrected, and crossref added by Geoffrey H. Morley, Mar 19 2010
Additional term added, initial term a(0)=1 deleted by Stuart E Anderson, Dec 26 2010
Upper bounds for terms a(31) to a(78), (all from Ian Gambini's thesis) added by Stuart E Anderson, Jan 20 2011
New bound for a(29)<=221, from Stuart E Anderson & Ed Pegg Jr, Jan 20 2011
a(29) confirmed as 221, from Stuart E Anderson, Ed Pegg Jr, and Stephen Johnson, Aug 22 2011
New bound for a(31)<=236, computed by Stephen Johnson in September 2011, updated by Stuart E Anderson, Oct 04 2011
a(30) from Stuart E Anderson and Lorenz Milla added by Geoffrey H. Morley, Jun 15 2013
a(31) and a(32) from Lorenz Milla and Stuart E Anderson, Oct 05 2013
For additional terms see the Ed Pegg link, also A006983.  N. J. A. Sloane, Jul 29 2020
a(33) to a(37) from J. B. Williams, added by Stuart E Anderson, Oct 27 2020


STATUS

approved



