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Is there an algorithm to find out all possible resistance values of series, parallel, and series-parallel arrangements given $n$ identical resistors, $R$? All of them must be used.

This might even extend to differently-valued resistors, but I'll just focus on identical resistors.

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Hmmm... what do you mean by "find out"? Do you want to construct them or count them? If you're after the number of series-parallel graphs with a given number of edges, you might want this: en.wikipedia.org/wiki/Series-parallel_networks_problem –  Douglas S. Stones Dec 17 '10 at 8:30
@Douglas: Count and construct. –  Kit Dec 17 '10 at 8:37
I am not sure what your definitions are (what a combination is, what your definition for when two combinations are the same) but if you could write down what you think the first few terms of the sequence is, you could look it up on the OEIS: oeis.org –  Qiaochu Yuan Dec 17 '10 at 8:39
@Qiaochu: I think I meant all resistor network arrangements, not some number series. I want to find out all resistance values possible with the resistors available to me. –  Kit Dec 17 '10 at 8:43

1 Answer 1

Following the Wikipedia link given by Douglas, and the references from there to OEIS, leads to this paper which seems to answer your question: Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).

The author points out that asking for the possible values of the resistance is not the same as asking for the total number of networks. The answer to the former question is claimed to be approximately $2.55^n$ (for large $n$, I presume).

[EDIT: A look in the paper reveals that the formula $2.55^n$ was simply obtained by straight-line fitting of the logarithm of the exact values for $6 \le n \le 16$, so (being a mathematician) I'm not quite convinced about its validity for large $n$...]

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The relevant OEIS links are oeis.org/A000084 and oeis.org/A048211. –  Hans Lundmark Dec 17 '10 at 11:32
"was simply obtained by straight-line fitting of the logarithm of the exact values" - so much for scientific rigor... :'( –  J. M. Dec 17 '10 at 14:11

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