Consider a network of
resistors
so that
may be connected in series or parallel with , may be connected in series or parallel with the network
consisting of
and ,
and so on. The resistance of two resistors in series is given by

(1)

and of two resistors in parallel by

(2)

The possible values for two resistors with resistances and are therefore

(3)

for three resistances ,
, and are

(4)

and so on. These are obviously all rational numbers, and the numbers of distinct arrangements for ,
2, ..., are 1, 2, 8, 46, 332, 2874, ... (OEIS A005840),
which also arises in a completely different context (Stanley 1991).

If the values are restricted to , then there are possible resistances for 1- resistors, ranging from a minimum of to a maximum of . Amazingly, the largest denominators 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, ..., , then the numbers of possible net resistances for 1, 2, ...
resistors are 1, 2, 8, 44, 298, 2350, ... (OEIS A051045).
The following table gives the values possible for small .

Amengual, A. "The Intriguing Properties of the Equivalent Resistances of
Equal Resistors Combined in Series and in Parallel." Amer. J. Phys.68,
175-179, 2000.Sloane, N. J. A. Sequences A000045/M0692,
A005840/M1872, and A051045
in "The On-Line Encyclopedia of Integer Sequences."Stanley,
R. P. "A Zonotope Associated with Graphical Degree Sequences." In
Applied
Geometry and Discrete Mathematics: The Victor Klee Festschrift (Ed. P. Gritzmann
and B. Sturmfels). Providence, RI: Amer. Math. Soc., pp. 555-570, 1991.