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 
.
 | possible resistances |
| 1 | 1 |
| 2 |  |
| 3 |  |
| 4 |  |
Equal Resistors Combined in Series and in Parallel." Amer. J. Phys. 68,
175-179, 2000.Sloane, N. J. A. Sequences