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 1-5 of 5 results.

A181340 Number of compound perfect squared squares of order n up to symmetries of the square and its squared subrectangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 16, 46, 143, 412, 941, 2788, 7941, 22413, 62273, 172330, 466508, 1239742, 3257378, 8430928
Offset: 1

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Author

Stuart E Anderson, Oct 13 2010, Oct 16 2010

Keywords

Comments

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. A squared rectangle is compound if it contains a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. - Geoffrey H. Morley, Oct 17 2012
The smallest perfect compound squared square was published by T. H. Willcocks in 1948, has 24 squares and has one rectangle as a sub-dissection; however, it was not until 1982 that A. J. W. Duijvestijn, P. J. Federico and P. Leeuw proved it to be the lowest-order example.
In 2010 Stuart Anderson and Ed Pegg Jr generated all 2-connected minimum degree 3 planar graphs up and including 29 edges, using B. D. McKay and G. Brinkmann's plantri software, then applied electrical node analysis to the graphs to obtain complete counts of compound perfect squares in orders 24, 25, 26, 27 and 28, along with all the members of each equivalence class of each compound square.
In 2011 S. E. Anderson and Stephen Johnson commenced order 29 CPSSs, and processed all plantri generated 2-connected minimum degree 3 planar graph embeddings with up to 15 vertices. This left the largest graph class, the 16 vertex class. In 2012 S. E. Anderson processed the remaining graphs, using the Amazon Elastic Cloud supercomputer and new software which he wrote to find a(29). - Stuart E Anderson, Nov 30 2012
In May 2013 Lorenz Milla and Stuart Anderson enumerated a(30) (CPSSs of order 30), using the same process and software as used on order 29 CPSSs, with the addition of a technique recommended by William Tutte in his writings which resulted in a 3x speed-up of the search for perfect squared squares by factoring the determinant of the Kirchhoff/discrete Laplacian matrix of a graph into a product 2fS, where f is a squarefree number and S is a square number. - Stuart E Anderson, May 26 2013
From June to September 2013, Lorenz Milla further optimized the process and software and completed the computation required to enumerate all CPSSs of order 31 and 32. A second run with enhanced software was undertaken by Milla and Anderson as there was a possibility some CPSSs could have been missed on the first run. The second run found nothing new or different and confirmed the result. - Stuart E Anderson, Sep 29 2013
In April 2014, Jim Williams wrote software and used it to complete the enumeration of CPSS orders 33, 34, 35 and 36. - Stuart E Anderson, May 02 2016
In August 2018, Jim Williams completed the enumeration of CPSS orders 37, 38 and 39. - Stuart E Anderson, Sep 17 2018.

Examples

			From _Geoffrey H. Morley_, Oct 17 2012 (Start):
See MathWorld link for an explanation of Bouwkamp code.
a(24)=1 because all four compound perfect squares of order 24 are equivalent up to symmetries. They have side 175. The Bouwkamp code for one of them is (81,56,38)(18,20)(55,16,3)(1,5,14)(4)(9)(39)(51,30)(29,31,64)(43,8)(35,2)(33). (End)

References

  • J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993. [Includes some compound perfect squares up to order 30.]
  • T. H. Willcocks, Problem 7795 & solution, Fairy Chess Review 7 (1948) 97, 106.

Links

Crossrefs

Cf. A217155 (counts symmetries of subrectangles as distinct).
Cf. A006983, A181735, A217152, A217153.

Extensions

Corrected last term from 142 to 143 to include cpss 1170C, added cross reference
Corrected last term from 143 to 144 to include cpss 1224d, incorrectly excluded as a duplicate in the initial count.
Corrected last term from 144 back to 143 after a recount from the original graphs established a bijection between exactly 948 non-isomorphic graphs and 948 isomers in 143 different CPSS arrangements. Gave usual bouwkampcode notation in examples. Removed redundant word "mathematically" from comments. - Stuart E Anderson, Jan 2012
Clarified the definition of 'number' in relation to the 'number' of compound squares, included the definition of 'perfect'. Excluded the trivial dissection from the sequence count. - Stuart E Anderson, May 2012
Definition corrected and offset changed to 1 by Geoffrey H. Morley, Oct 17 2012
a(29) added by Stuart E Anderson, Nov 30 2012
a(30) added by Stuart E Anderson, May 26 2013
a(31)-a(32) added by Stuart E Anderson, Sep 29 2013
a(33)-a(36), enumeration of these orders was completed by Jim Williams in 2014, added by Stuart E Anderson, May 02 2016
a(37)-a(39), enumeration of these orders was completed by Jim Williams in 2018, added by Stuart E Anderson, Sep 17 2018

A217153 Number of nontrivially compound perfect squared rectangles of order n up to symmetries of the rectangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 48, 264, 1256, 5396, 22540, 92060, 370788
Offset: 1

Views

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

Comments

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, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

Links

Crossrefs

Cf. A217152 (counts symmetries of squared subrectangles as equivalent).
Cf. A002839, A181340, A217154, A217155.

Extensions

a(19) and a(20) corrected (thanks to Stuart E Anderson's computations which show I misinterpreted Gambini's counts) by Geoffrey H. Morley, Oct 12 2012

A217155 Number of compound perfect squared squares of order n up to symmetries of the square.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 12, 100, 220, 948, 2308, 5668, 17351, 52196, 150669, 429458, 1206181, 3337989, 8961794, 23989218, 62894424
Offset: 1

Views

Author

Geoffrey H. Morley, Sep 27 2012

Keywords

Comments

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. A squared rectangle is compound if it contains a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares.
The terms up to a(26) were first published by Gambini (1999) but included no new squarings neither counted by Duijvestijn, Federico and Leeuw (1982) nor in Skinner's book (1993). In 2010 Anderson and Pegg used plantri and Anderson's programs to confirm Gambini's counts and to find a(27) and a(28).
In 2011, S. E. Anderson and Stephen Johnson commenced order 29 CPSSs, and processed all plantri generated 2-connected minimum degree 3 planar graph embeddings with up to 15 vertices. This left the largest graph class, the 16 vertex class. In 2012, S. E. Anderson processed the remaining graphs, using the Amazon Elastic Cloud supercomputer and new software which he wrote to find a(29). - Stuart E Anderson, Nov 30 2012
In May 2013, Lorenz Milla and Stuart Anderson enumerated a(30) (CPSSs of order 30), using the same process and software as used on order 29 CPSSs, with the addition of a technique recommended by William Tutte in his writings which resulted in a 3x speed up of the search for perfect squared squares by factoring the determinant of the Kirchhoff/discrete Laplacian matrix of a graph into a product 2fS, where f is a squarefree number and S is a square number. - Stuart E Anderson, May 26 2013
From June to September 2013, Lorenz Milla further optimized the process and software and completed the computation required to enumerate all CPSSs of order 31 and 32. A second run with enhanced software was undertaken by Milla and Anderson as there was a possibility some CPSSs could have been missed on the first run. The second run found nothing new or different and confirmed the result. - Stuart E Anderson Sep 29 2013
In April 2014, Jim Williams wrote software and enumerated all CPSSs in orders 33, 34, 35 and 36. - Stuart E Anderson May 02 2016
In August 2018, Jim Williams completed the enumeration of all CPSSs and CPSS isomers in orders 37, 38 and 39. - Stuart E Anderson, Sep 17 2018

Examples

			See MathWorld link for an explanation of Bouwkamp code.
a(24)=4 because the compound perfect squares of order 24 comprise the one with side 175 and Bouwkamp code (81,56,38) (18,20) (55,16,3) (1,5,14) (4) (9) (39) (51,30) (29,31,64) (43,8) (35,2) (33) and three others from the other symmetries of the squared subrectangle.

References

  • J. D. Skinner II, Squared Squares: Who's Who & What's What, published by the author, 1993. [Includes some compound perfect squares up to order 30.]

Links

Crossrefs

Cf. A181340 (counts symmetries of squared subrectangles as equivalent).
Cf. A006983, A217152, A217153, A217156.

Extensions

a(29) from Stuart E Anderson, Nov 30 2012
a(30) from Stuart E Anderson, May 26 2013
a(31)-a(32) from Stuart E Anderson, Sep 29 2013
Minor edits by Jon E. Schoenfield, Feb 15 2014
a(33)-a(36) from Stuart E Anderson, May 02 2016
a(37)-a(39) from Stuart E Anderson, Sep 17 2018

A110148 Number of perfect squared rectangles of order n up to symmetries of the rectangle and of its subrectangles if any.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 38, 127, 408, 1375, 4783, 16645, 58059, 203808, 722575
Offset: 1

Views

Author

Tanya Khovanova, Feb 18 2007

Keywords

Comments

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. [Geoffrey H. Morley, Oct 12 2012]

Links

  • C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(12) and an incorrect value for a(13) on p. 1178.]
  • C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625). [Correct terms up to a(13) on p. 1299.]
  • I. M. Yaglom, How to dissect a square? (in Russian), Nauka, Moscow, 1968. In djvu format (1.7M), also as this pdf (9.5M). [Terms up to a(13) on pp. 26-7.]
  • Index entries for squared rectangles
  • Index entries for squared squares

Crossrefs

Cf. A217154 (counts symmetries of any subrectangles as distinct).
Cf. A181735, A217153, A217156.

Formula

a(n) = A002839(n) + A217152(n) + A217374(n). - Geoffrey H. Morley, Oct 12 2012
a(n) = a(n-1) + A002839(n) + A002839(n-1) + A217152(n) + A217152(n-1). - Geoffrey H. Morley, Oct 12 2012

Extensions

Definition corrected and a(14)-a(19) added by Geoffrey H. Morley, Oct 12 2012

A217374 Number of trivially compound perfect squared rectangles of order n up to symmetries of the rectangle and its subrectangles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 16, 60, 194, 622, 2128, 7438, 25852, 90266, 317350, 1127800
Offset: 1

Views

Author

Geoffrey H. Morley, Oct 02 2012

Keywords

Comments

A squared rectangle 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, compound if it does, and trivially compound if a constituent square has the same side length as a side of the squared rectangle under consideration.

Links

  • C. J. Bouwkamp, On the dissection of rectangles into squares (Papers I-III), Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, Paper I, 49 (1946), 1176-1188 (=Indagationes Math., v. 8 (1946), 724-736); Paper II, 50 (1947), 58-71 (=Indagationes Math., v. 9 (1947), 43-56); Paper III, 50 (1947), 72-78 (=Indagationes Math., v. 9 (1947), 57-63). [Paper I has terms up to a(13) on p. 1178.]
  • C. J. Bouwkamp, On the construction of simple perfect squared squares, Koninklijke Nederlandsche Akademie van Wetenschappen, Proc., Ser. A, 50 (1947), 1296-1299 (=Indagationes Math., v. 9 (1947), 622-625).
  • Index entries for squared rectangles

Crossrefs

Cf. A217375 (counts symmetries of squared subrectangles as distinct).
Cf. A110148.

Formula

a(n) = a(n-1) + 2*A002839(n-1) + 2*A217152(n-1).

Extensions

a(20) corrected by Geoffrey H. Morley, Oct 12 2012
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