Resistor Network

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

REFERENCES:

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.