cp's OEIS Frontend

Karnofsky (2004, p. 5): "[...] if some circuit has resistance m/n then some other circuit likely has n/m. In fact, for 9 or fewer resistors, this symmetry is perfect. However, for 10 resistors the following values are achieved, but not their inverses: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67". That means, that T(m,n) = T(n,m), if T(m,n) <= 9.

This starts with the values of A113881, but the Karnofsky comment says that T(n,m) is not symmetric, whereas the count of tiles in A113881 is. - R. J. Mathar, Nov 06 2020

The first difference where T(m,n) = T(n,m), but differs from the corresponding entry of A113881 occurs for (n,m) = (154,167) and (n,m) = (167,154), both representable by networks with non-planar graphs of 11 resistors, whereas A113881 counts 12 tiles. See Pfoertner link for illustration of more differences. - Hugo Pfoertner, Nov 13 2020

A connected multigraph G with a selected pair P of nodes can be used to represent a resistor network. The edges represent resistors, and the total resistance is measured between the selected nodes. It is possible to construct complex networks using only serial or parallel combinations, but the more nodes and edges are involved, the more networks of a different kind can be found. They cannot be decomposed into serial/parallel elements. The sequence is on page 2 of the paper describing the computation of A180414 (see the Joel Karnofsky link).

Karnofsky claims that he systematically increased the number of edges by three basic operations, C, D, and E, defined in A338999, i.e., he claims to have counted the CDE-descendants of the simplest h-graph (the "bridge," see the example section). Numbers given in his paper are 1, 5, 37, 226, 1460, 9235, which is slightly off (see A339386). The difference seems to stem from the "dangling parts," as he calls them in his "addendum," so they don't affect the computation of different resistances in A180414. - Rainer Rosenthal, Dec 02 2020

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.

This sequence counts the CDE-descendants of a single edge A-Z.

[C]onnect: different nodes {P,Q} != {A,Z} may form a new edge P-Q.

[D]issect: any edge P-Q may be dissected into P-M-Q with a new node M.

[E]xtend: any node P not in {A,Z} may form a new edge P-Q with a new node Q.

These basic operations were motivated by A338487, which seemed to count the CDE-descendants of K_4 with edge A-Z removed.

This sequence is an analog of A338861. Equality a(n) = A338861(n) holds for small n only, see example.

Let V_s denote the specific voltage, i.e., the lowest integer voltage, which induces integer currents everywhere in the network. Denote by A_s the specific current, i.e., the corresponding total current.

A planar network with n unit resistors corresponds to a squared rectangle with height V_s and width A_s. The electrical power V_s*A_s therefore equals the area of that rectangle. In the historical overview (Stuart Anderson link) A_s is called complexity.

The corresponding rectangle tiling provides the optimal power rating of the 1 ohm resistors with respect to the specific voltage V_s and current A_s. See the picture From_Quilt_to_Net in the link section, which also provides insight in the "mysterious" correspondence between rectangle tilings and electric networks. For non-planar nets the idea of rectangle tilings can be widened to 'Cartesian squarings'. A Cartesian squaring is the dissection of the product P X Q of two finite sets into 'squaresets', i.e., sets A X B with A subset of P and B subset of Q, and card(A) = card(B). - Rainer Rosenthal, Dec 14 2022

Take the set SetA337517(n) of resistances, counted by A337517. For each resistance R multiply numerator and denominator. Conjecture: a(n) is the maximum of all these products. The reason is that common factors of V_s and A_s are quite rare (see the beautiful exceptional example with 21 resistors).

The relation of this sequence to A340920 is the analog of the relation of A180414 to A337517.