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In "addendum" J. Karnofsky stated the value a(15) = 1064833. In contrast to the terms up to and including a(14), which could all be confirmed, an independent calculation based on a list of 3-connected simple graphs resulted in the corrected value a(15) = 1064835. - Hugo Pfoertner, Dec 06 2020

See A337517 for the number of different resistances that can be obtained by combining /exactly/ n one-ohm resistors. The method used by Andrew Howroyd (see his program in the link section) uses 3-connected graphs with one edge (the 'battery edge') removed. - Rainer Rosenthal, Feb 07 2021

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

Connected multigraphs rooted at vertices A and Z can be considered as resistor networks with 1-ohm-resistors per edge and total resistance measured between A and Z.

The networks counted here are a subset of the networks counted by A338999. Due to the 3-connectedness with respect to the two distinguished vertices none of these resistor networks is a parallel combination.

For a resistor network to be effective, one has to avoid dead ends. A dead end is a subgraph which becomes isolated from the distinguished vertices by the removal of one of its vertices. Since the multigraph is 2-connected, there are no dead ends. Another consequence of the 2-connectedness is, that the resistor network is not a series combination (like Fig. 5 in the example).

Karnofsky states in the addendum: "A graph has no dangling parts that don't affect the effective resistance if and only if it is 2-connected. A new idea is that the essential graphs to generate are 2-connected ones with minimal order (edges per node) 3". In this sequence there is no restriction w.r.t. the degree.

So the networks with n resistors counted by a(n) are neither parallel nor serial combinations, but they form networks which Karnofsky described as "h-graphs" (see A338487). The number of different resistance values is the same as for the respective networks in A338487.

Let us write Net = (E,V,A,Z) to denote the network consisting of E = set of edges, V = set of vertices, A and Z the distinguished vertices in V. Two networks (E1,V1,A1,Z1) and (E2,V2,A2,Z2) are counted only once, if there exists a bijection b: V1 -> V2 which sends E1 to E2 and {A1,Z1} to {A2,Z2}. Thus symmetrical networks w.r.t. A and Z are counted only once.

Variant of A339123, treating the distinguished points as not interchangeable.

Graphs that are 2-connected also have no decomposition into series components, so the graphs enumerated by this sequence are the minimal subset of oriented networks which when combined in series and parallel produce all possible networks with a source and a sink and in which every edge lies on a path between the source and the sink.