cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005670 Mrs. Perkins's quilt: smallest coprime dissection of n X n square.

Original entry on oeis.org

1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

The problem is to dissect an n X n square into smaller integer squares, the GCD of whose sides is 1, using the smallest number of squares. The GCD condition excludes dissecting a 6 X 6 into four 3 X 3 squares.
The name "Mrs Perkins's Quilt" comes from a problem in one of Dudeney's books, wherein he gives the answer for n = 13. I gave the answers for low n and an upper bound of order n^(1/3) for general n, which Trustrum improved to order log(n). There's an obvious logarithmic lower bound. - J. H. Conway, Oct 11 2003
All entries shown are known to be correct - see Wynn, 2013. - N. J. A. Sloane, Nov 29 2013

Examples

			Illustrating a(7) = 9: a dissection of a 7 X 7 square into 9 pieces, courtesy of _Ed Pegg Jr_:
.___.___.___.___.___.___.___
|...........|.......|.......|
|...........|.......|.......|
|...........|.......|.......|
|...........|___.___|___.___|
|...........|...|...|.......|
|___.___.___|___|___|.......|
|...............|...|.......|
|...............|___|___.___|
|...............|...........|
|...............|...........|
|...............|...........|
|...............|...........|
|...............|...........|
|___.___.___.___|___.___.___|
The Duijvestijn code for this is {{3,2,2},{1,1,2},{4,1},{3}}
Solutions for n = 1..10: 1 {{1}}
2 {{1, 1}, {1, 1}}
3 {{2, 1}, {1}, {1, 1, 1}}
4 {{2, 2}, {2, 1, 1}, {1, 1}}
5 {{3, 2}, {1, 1}, {2, 1, 2}, {1}}
6 {{3, 3}, {3, 2, 1}, {1}, {1, 1, 1}}
7 {{4, 3}, {1, 2}, {3, 1, 1}, {2, 2}}
8 {{4, 4}, {4, 2, 2}, {2, 1, 1}, {1, 1}}
9 {{5, 4}, {1, 1, 2}, {4, 2, 1}, {3}, {2}}
10 {{5, 5}, {5, 3, 2}, {1, 1}, {2, 1, 2}, {1}}

References

  • H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C3.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005842, A089046, A089047.

Extensions

b-file from Wynn 2013, added by N. J. A. Sloane, Nov 29 2013

A089047 Edge length of largest square dissectable into up to n squares in Mrs. Perkins's quilt problem.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 23, 29, 41, 53, 70, 91, 126, 158, 216, 276, 386, 488, 675, 866, 1179, 1544, 2136, 2739, 3755, 4988, 6443
Offset: 1

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Author

R. K. Guy, Dec 03 2003

Keywords

Comments

An inverse to A005670.
More precisely, a(n) = greatest k such that A005670(k) <= n. - Peter Munn, Mar 13 2018
It is not clear which terms have been proved to be correct and which are just conjectures. - Geoffrey H. Morley, Sep 07 2012; N. J. A. Sloane, Jul 06 2017
Terms up to and including a(18) have been proved correct by Ed Wynn (2013). - Stuart E Anderson, Sep 16 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
Further best known terms are 8568, 11357, 14877, 19594, 26697, 34632. - Ed Pegg Jr, Jul 06 2017
A290821 is the equivalent sequence for equilateral triangles. - Peter Munn, Mar 06 2018

Links

Crossrefs

Cf. A005670, A014529, A018835, A089046, A211302, A290821.

Extensions

More terms from Ed Pegg Jr, Dec 03 2003
Corrected and extended by Ed Pegg Jr, Apr 18 2010
Duplicate a(6) deleted and a(22)-a(26) revised (from Ed Pegg Jr, Jun 15 2010) by Geoffrey H. Morley, Sep 07 2012
Conjectured terms have been extended up to a(44), based on simple squared square enumeration, by Duijvestijn, Skinner, Anderson, Pegg, Johnson, Milla and Williams. - Stuart E Anderson, Sep 16 2013
a(33) and further terms added by Ed Pegg Jr, Jul 06 2017
Name edited by Peter Munn, Mar 14 2018

A220164 Number of simple squared squares of order n up to symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 5, 15, 19, 57, 72, 275, 499, 1778, 3705, 11318, 24525, 65906, 135599, 333938, 687969, 1681759, 3652677
Offset: 1

Views

Author

Stuart E Anderson, Dec 06 2012

Keywords

Comments

A squared rectangle is a rectangle dissected into a finite number, two or more, of squares, called the elements of the dissection. If no two of these squares have the same size the squared rectangle is called perfect, otherwise it is imperfect. The order of a squared rectangle is the number of constituent squares. The case in which the squared rectangle is itself a square is called a squared square. The dissection is simple if it contains no smaller squared rectangle, otherwise it is compound. This sequence counts both perfect and imperfect simple squared squares up to symmetry.

References

Links

Crossrefs

Cf. A006983, A002962, A217156, A089046.

Formula

a(n) = A006983(n) + A002962(n).

Extensions

a(13)-a(29) from Stuart E Anderson, Dec 07 2012
Clarified some definitions in comments and added a(30) - Stuart E Anderson, Jun 03 2013
a(31), a(32) added by Stuart E Anderson, Sep 30 2013
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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