cp's OEIS Frontend

One can view a circuit with n unit resistors as a multigraph G with n edges and a pair P of distinguished nodes. Every edge of the graph must be contained in a path connecting the two distinguished nodes.

In case n > 0, a(n) counts all resistances R(G, P), which are rational numbers by Kirchhoff's laws. In case n = 0, the graph G consists of only two pair P nodes, and there is only one resistance: oo = infinity; so a(0) = 1. In the OEIS, there are already sequences that count the possible resistances of circuits of certain types (for the definitions see A337516).

OEIS | type | 1 2 3 4 5 6 7 8 9 10 11 12 13

---------+------+--------------------------------------------------------------

A048211 | SP | [1] 2 4 9 22 53 131 337 869 2213 5691 14517 37017

A174283 | SPB | 1 2 4 9 23 [57] 151 415 1157 3191 8687 23199 61677

A337516 | SPBF | 1 2 4 9 23 57 151 [421] 1202 3397 9498 25970 70005

A337517 | all | 1 2 4 9 23 57 151 [427] 1263 3823 11724 36048 110953

The table shows the number of different resistances, which grows with the complexity of the circuits. Values in square brackets mark the beginning of the newly explored range. Values a(n) up to n = 7 are fully classified, and have one of the given types, i.e., they can be computed by the functions Ser(), Par(), Bri(), and Frk() defined in A337516. For a(n), n >= 8, the theory in A180414 has to be applied.

Note: The 'set counted by A180414(n)' is the union of all 'sets counted by A337517(k) for k = 0 .. n'.

Admissible networks (G, P) are those defined in the Karnofsky paper (A180414).

This sequence is a variation on A048211, which uses only series and parallel combinations. Since a bridge circuit requires minimum of five resistances the first four terms coincide. For the definition of "bridge" see A337516.

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176500 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 10 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 25 2021

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176502 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, Jul 12 2019]

Farey(n) = A005728(n). - Franklin T. Adams-Watters, May 12 2010

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 11 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021

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

Cumulative sequence based on A337516.