

A089046


Least edgelength of a square dissectable into at least n squares in the Mrs. Perkins's quilt problem.


4



1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 10, 14, 18, 24, 30, 40, 54, 71, 92, 121, 155, 210, 266, 360, 476, 642, 833, 1117, 1485, 1967, 2595, 3465, 4534, 5995
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OFFSET

1,2


COMMENTS

An inverse to A005670.
More precisely, a(n) = least k such that A005670(k) >= n.  Peter Munn, Mar 14 2018
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
n <= 15 (and possibly 16) proved minimal by J. H. Conway (Conway, J. H. "Re: [mathfun] Mrs. Perkins Quilt  Orders 89, 90 improved over UPIG." mathfun mailing list. October 10, 2003.). The conjectures are best currently known values of a(n) for n > 16.  Stuart E Anderson, Apr 21 2013
A089046 and A089047 are almost certainly correct up to 5000.  Ed Pegg Jr, Jul 06 2017
Deleted terms above 5000.  N. J. A. Sloane, Jul 06 2017
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


REFERENCES

H. T. Croft, K. J. Falconer, and R. K. Guy, Section C3 in Unsolved Problems in Geometry, New York: Springer, 1991.
M. Gardner, "Mrs. Perkins's Quilt and Other SquarePacking Problems," Mathematical Carnival, New York: Vintage, 1977.


LINKS

Table of n, a(n) for n=1..34.
Stuart E. Anderson, Mrs Perkins's Quilts
J. H. Conway, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 60, 363368, 1964.
Ed Pegg, Jr., Mrs. Perkin's Quilts
Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts (Wolfram Demonstrations Project)
Ed Pegg Jr., Richard K. Guy, Mrs. Perkins's Quilts Notebook source code
G. B. Trustrum, Mrs. Perkins's Quilt, Proc. Cambridge Phil. Soc. 61, 711, 1965.
Eric W. Weisstein's World of Mathematics, Mrs. Perkins's Quilt
Ed Wynn, Exhaustive generation of Mrs Perkins's quilt square dissections for low orders, arXiv:1308.5420 [math.CO], 20132014.


CROSSREFS

Cf. A005670, A089046, A089047.
Sequence in context: A016085 A018122 A074732 * A054911 A137267 A123576
Adjacent sequences: A089043 A089044 A089045 * A089047 A089048 A089049


KEYWORD

nonn,hard,more


AUTHOR

R. K. Guy, Dec 03 2003


EXTENSIONS

More terms from Ed Pegg Jr, Dec 03 2003
Corrected and extended by Ed Pegg Jr, Apr 18 2010
a(24)a(27) (from Ed Pegg Jr, Jun 15 2010) added by Geoffrey H. Morley, Aug 29 2012
Added a(28)a(30) from Stuart E Anderson, Mov 22 2012
Confirmed a(30) as best known, added a(31) as best known.  Stuart E Anderson, Apr 21 2013
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
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
a(1)a(19) confirmed by Ed Wynn, 2013.  N. J. A. Sloane, Nov 29 2013
a(29) corrected and further terms added by Ed Pegg Jr, Jul 06 2017


STATUS

approved



