The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel

TL;DR

This paper introduces an analytical method using Farey sequences and Fibonacci numbers to determine bounds of equivalent resistances for n equal resistors in series and parallel, extending to complex circuit configurations.

Contribution

It presents a novel analytical approach for bounding the set of equivalent resistances using Farey sequences, surpassing previous computational limitations.

Findings

Approximate growth formula A(n) ~ 2.55^n matches data up to n=22.

Strict upper bound A(n) ~ 2.618^n established.

Farey sequence method applicable to complex circuit configurations.

Abstract

The order of the set of equivalent resistances, A(n) of n equal resistors combined in series and in parallel has been traditionally addressed computationally, for n up to 22. For larger n there have been constraints of computer memory. Here, we present an analytical approach using the Farey sequence with Fibonacci numbers as its argument. The approximate formula, A(n) ~ 2.55^n, obtained from the computational data up to n = 22 is consistent with the strict upper bound, A(n) ~ 2.618^n, presented here. It is further shown that the Farey sequence approach, developed for the A(n) is applicable to configurations other than the series and/or parallel, namely the bridge circuits and non-planar circuits. Expressions describing set theoretic relations among the sets A(n) are presented in detail. For completeness, programs to generate the various integer sequences occurring in this study, using the symbolic computer language MATHEMATCA, are also presented.