cp's OEIS Frontend

A squared rectangle (which may be a square) is a rectangle dissected into a finite number of two or more squares. If no two squares have the same size, the squared rectangle is perfect. A squared rectangle is simple if it does not contain a smaller squared rectangle. The order of a squared rectangle is the number of constituent squares. - Geoffrey H. Morley, Oct 17 2012

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, and compound if it does. - Geoffrey H. Morley, Oct 17 2012

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.

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

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.

It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".

In 1983, A. Augusteijn and A. J. W. Duijvestijn described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")

For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).

In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.

In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

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.

The graphs are exactly 2-connected, not at least 2-connected. The graphs were enumerated using plantri (by B.D. McKay & G. Brinkmann) for the purpose of finding compound perfect squared squares.

The graphs are 2-connected, but not 3-connected. The graphs were enumerated using plantri (by B.D. McKay & G. Brinkmann) for the purpose of finding compound perfect squared squares. If all graphs with n edges are generated then all compound squares in order n-1 can be obtained from them. Graphs with minimum degree at least 3 are also called homeomorphically irreducible.

The isomer count of a compound perfect squared square (CPSS) is the number of ways its squared subrectangle and constituent squares can be arranged, up to symmetry of the CPSS. A squared square is perfect if none of its constituent squares are the same size. A squared square is compound if it contains a smaller squared subrectangle. Note that the squared subrectangle can be a squared square. Specific concrete examples of CPSSs with isomer counts under 100 of 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 28, 31, 32, 35, 36, 39, 40, 47, 48, 56, 60, 63, 64, 68, 72, 76, 80, 88 and 96 exist. Geometric constructions based on a suitable pair of perfect squared rectangles each with up to 4 isomers suggests additional isomer counts up to 100 of 14, 21, 22, 33, 42, 44, 66 and 99, but no actual examples are known. As the number of squares in a squared square - the order - increases new arrangements appear. It is conjectured that expected CPSS subrectangle isomer arrangements will eventually appear if the order is high enough.

The term a(6)=14 is based on a theoretical construction, not on known or existing CPSSs. These terms have been included to distinguish the sequence from others. Considering all the ways two or more subrectangles can be arranged within a CPSS it does not appear possible for a CPSS with 5, 6, 9, 10 or 13 isomers to exist but even this much has not been proved.