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).
