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A089046 Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem. 4


%S 1,2,2,2,3,3,4,5,6,8,10,14,18,24,30,40,54,71,92,121,155,210,266,360,

%T 476,642,833,1117,1485,1967,2595,3465,4534,5995

%N Least edge-length of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem.

%C An inverse to A005670.

%C More precisely, a(n) = least k such that A005670(k) >= n. - _Peter Munn_, Mar 14 2018

%C It is not clear which terms have been proved to be correct and which are just conjectures. - _Geoffrey H. Morley_, Aug 29 2012; _N. J. A. Sloane_, Jul 06 2017

%C n <= 15 (and possibly 16) proved minimal by _J. H. Conway_ (Conway, J. H. "Re: [math-fun] Mrs. Perkins Quilt - Orders 89, 90 improved over UPIG." math-fun mailing list. October 10, 2003.). The conjectures are best currently known values of a(n) for n > 16. - _Stuart E Anderson_, Apr 21 2013

%C A089046 and A089047 are almost certainly correct up to 5000. - _Ed Pegg Jr_, Jul 06 2017

%C Deleted terms above 5000. - _N. J. A. Sloane_, Jul 06 2017

%C Upper bounds for the next terms in the sequence (which may well be the true values) are 7907, 10293, 13505, 17785, 23239, 31035, 39571, ... - _Ed Pegg Jr_, Jul 06 2017

%D H. T. Croft, K. J. Falconer, and R. K. Guy, Section C3 in Unsolved Problems in Geometry, New York: Springer, 1991.

%D M. Gardner, "Mrs. Perkins's Quilt and Other Square-Packing Problems," Mathematical Carnival, New York: Vintage, 1977.

%H Stuart E. Anderson, <a href="http://www.squaring.net/quilts/mrs-perkins-quilts.html">Mrs Perkins's Quilts</a>

%H J. H. Conway, <a href="http://dx.doi.org/10.1017/S0305004100037877">Mrs. Perkins's Quilt</a>, Proc. Cambridge Phil. Soc. 60, 363-368, 1964.

%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/perkinsbestquilts.txt">Mrs. Perkin's Quilts</a>

%H Ed Pegg Jr., Richard K. Guy, <a href="http://demonstrations.wolfram.com/MrsPerkinssQuilts">Mrs. Perkins's Quilts</a> (Wolfram Demonstrations Project)

%H Ed Pegg Jr., Richard K. Guy, <a href="http://demonstrations.wolfram.com/MrsPerkinssQuilts/MrsPerkinssQuilts-source.nb">Mrs. Perkins's Quilts Notebook source code</a>

%H G. B. Trustrum, <a href="https://doi.org/10.1017/S0305004100038573">Mrs. Perkins's Quilt</a>, Proc. Cambridge Phil. Soc. 61, 7-11, 1965.

%H Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MrsPerkinssQuilt.html">Mrs. Perkins's Quilt</a>

%H Ed Wynn, <a href="http://arxiv.org/abs/1308.5420">Exhaustive generation of Mrs Perkins's quilt square dissections for low orders</a>, arXiv:1308.5420 [math.CO], 2013-2014.

%Y Cf. A005670, A089046, A089047.

%K nonn,hard,more

%O 1,2

%A _R. K. Guy_, Dec 03 2003

%E More terms from _Ed Pegg Jr_, Dec 03 2003

%E Corrected and extended by _Ed Pegg Jr_, Apr 18 2010

%E a(24)-a(27) (from _Ed Pegg Jr_, Jun 15 2010) added by _Geoffrey H. Morley_, Aug 29 2012

%E Added a(28)-a(30) from _Stuart E Anderson_, Mov 22 2012

%E Confirmed a(30) as best known, added a(31) as best known. - _Stuart E Anderson_, Apr 21 2013

%E Using James Williams recent discoveries of 15 million simple perfect squared squares in orders 31 to 44 I was able to extend the sequence of best currently known values for optimal quilts from a(32) to a(44). - _Stuart E Anderson_, Apr 21 2013

%E Using Anderson and Milla's enumeration of order 31 and 32 perfect squared squares, improved conjectures for a(32) and a(33) were obtained - _Stuart E Anderson_, Sep 16 2013

%E a(1)-a(19) confirmed by Ed Wynn, 2013. - _N. J. A. Sloane_, Nov 29 2013

%E a(29) corrected and further terms added by _Ed Pegg Jr_, Jul 06 2017

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Last modified January 22 14:24 EST 2019. Contains 319364 sequences. (Running on oeis4.)