%I #77 May 23 2023 04:16:41

%S 1,1,2,4,9,23,57,151,427,1263,3823,11724,36048,110953,342079,1064468,

%T 3341067,10583564,33727683,107931482,346615834

%N a(n) is the number of distinct resistances that can be produced from a circuit with exactly n unit resistors.

%C 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.

%C 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).

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

%C ---------+------+--------------------------------------------------------------

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

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

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

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

%C 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.

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance">Electrical resistance and conductance</a>

%H <a href="/index/Res#resistances">Index to sequences related to resistances</a>.

%e For a(n) up to n = 7 see the above mentioned sequences.

%Y Cf. A048211, A180414, A174283, A337516, A338197.

%K nonn,hard,more,nice

%O 0,3

%A _Rainer Rosenthal_ and _Hugo Pfoertner_, Oct 29 2020

%E a(8)-a(14) from _Andrew Howroyd_, Oct 31 2020

%E a(15)-a(16) from _Hugo Pfoertner_, Dec 06 2020

%E a(17) from _Hugo Pfoertner_, Dec 09 2020

%E a(18) from _Hugo Pfoertner_, Apr 09 2021

%E a(19) from _Zhao Hui Du_, May 15 2023

%E a(20) from _Zhao Hui Du_, May 23 2023