A174286 Number of distinct resistances that can be produced using at most n equal resistors in series and/or parallel, confined to the five arms (four arms and the diagonal) of a bridge configuration. Since the bridge requires a minimum of five resistors, the first four terms are zero.
0, 0, 0, 0, 1, 3, 19, 75, 291, 985, 3011, 8659, 24319, 65899, 176591, 464451, 1211185
Offset: 1
Examples
Example 1: Five equal unit resistors. Each arm of the bridge has one unit resistor, leading to an equivalent resistance of 1; so the set is {1} and its order is 1. Example 2: Six equal unit resistors. Four arms have one unit resistor each and the fifth arm has two unit resistors. Two resistors in the same arm, when combined in series and parallel result in 2 and 1/2 respectively (corresponding to 2: {1/2, 2} in A048211). The set {1/2, 2}, in the diagonal results in {1}. Set {1/2, 2} in any of the four arms results in {11/13, 13/11}. Consequently, with six equal resistors, we have the set {11/13, 1, 13/11}, whose order is 3. Union of the previous terms is {1} and the union with these three is again {11/13, 1, 13/11}. So the terms for five and six resistors are 1 and 3 respectively.
Links
- Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). Digital Object Identifier (DOI): 10.1119/1.19396.
- Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
- Rainer Rosenthal, Maple programs SetA174285 and SetA174286
- Marx Stampfli, Bridged graphs, circuits and Fibonacci numbers, Applied Mathematics and Computation, Volume 302, 1 June 2017, Pages 68-79.
- Index to sequences related to resistances.
Crossrefs
Extensions
From Stampfli's paper, a(8) corrected and a(9)-a(12) added by Eric M. Schmidt, Sep 09 2017
Name edited by Eric M. Schmidt, Sep 09 2017
a(13)-a(17) added by Rainer Rosenthal, Feb 05 2021