Mathematical properties of resistor networks

We examine the sets of equivalent resistances formed by combining n number of equal resistors in series and parallel. It is seen that there is no known formula for the order of the sets thus formed. So, we obtain a set of inequalities for the lower and upper bounds for the order of the sets. Combinatorial arguments are used for obtaining the lower and upper bounds of the order of the sets. It is shown that a stricter upper bound can be obtained using the Fibonacci numbers and the Farey sequences. The analytical results are compared with the numerical results.