Resistor Network

A resistor network is a network of resistors. Consider ideal resistors with positive finite resistances . In an incremental construction, choose an ordering of the resistors and connect each new resistor in series or parallel with the entire preceding network. A series connection has a common junction with no additional branch, while a parallel connection joins the two pairs of terminals. The resistance of two resistors in series is given by

(1)

and of two resistors in parallel by

(2)

The possible values for three resistances , , and are

(3)

and so on. The resistances are rational functions of the individual resistances and are rational numbers when all the individual resistances are rational. Allowing any ordering, the numbers of distinct incremental arrangements, up to associativity and commutativity of the series and parallel connections, for , 2, ... are 1, 2, 8, 46, 332, 2874, ... (OEIS A005840). This sequence also arises in a completely different context (Stanley 1991, p. 566). Combining larger subnetworks in series or parallel instead of adding a single resistor at each step gives additional arrangements. For arbitrary series-parallel networks with distinct labelled resistors, the numbers of arrangements, identifying associative regroupings and reorderings in both series and parallel, are 1, 2, 8, 52, 472, 5504, ... (OEIS A006351). These count labelled structures, not distinct numerical resistance values.

If the values are restricted to , then this incremental construction gives distinct resistances for 1- resistors, ranging from a minimum of to a maximum of . Amazingly, the largest denominators in lowest terms for , 2, ... are 1, 2, 3, 5, 8, 13, 21, ..., which are immediately recognizable as the Fibonacci numbers (OEIS A000045). The following table gives the values possible for small .

	possible resistances
1	1
2
3
4

If the resistors are given the values 1, 2, ..., and may be inserted in any order, then the numbers of distinct net resistances for 1, 2, ... resistors are 1, 2, 8, 44, 298, 2350, ... (OEIS A051045). The following table gives the values for the fixed insertion order 1, 2, ..., for small .

	possible resistances
1	1
2
3
4

The restriction to adding one resistor at a time excludes some series-parallel resistor networks. If arbitrary series-parallel combinations are allowed, four unit resistors have nine distinct equivalent resistances, including 1 from two series-connected pairs placed in parallel. More generally, the numbers of distinct equivalent resistances obtainable from exactly , 2, ... unit resistors by arbitrary series-parallel combinations are 1, 2, 4, 9, 22, 53, 131, ... (OEIS A048211). For resistors of values 1, 2, 3, and 4, there are 48 distinct equivalent resistances rather than the 44 obtained by adding one resistor at a time.