A174283 Number of distinct resistances that can be produced using n equal resistors in, series, parallel and/or bridge configurations.
1, 2, 4, 9, 23, 57, 151, 415, 1157, 3191, 8687, 23199, 61677, 163257, 432541, 1146671, 3039829
Offset: 1
Examples
Example 1: Five 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. Thus a(5) = A048211(5) + 1 = 23.
Example 2: Six unit resistors: a bridge with 6 resistors yields A174285(6) = 3 different resistances and the series parallel combinations give A048211(6) = 53 resistances, but resistance 1 is counted twice. The union of the forementioned resistances has cardinality 53+3-1 = 55. There are two more circuits to be considered: the bridge with five unit resistors and the sixth unit resistor either in parallel (value 1/2) or in series (value 2). Both values 1/2 and 2 are not counted by A048211(6) or A174285(6), so the total is 55 + 2 = 57. - _Rainer Rosenthal_, Oct 25 2020
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).
- 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).
- Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162.
- Sameen Ahmed Khan, Beginning to Count the Number of Equivalent Resistances, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 1-7, 2016.
- Hugo Pfoertner, Increase of number of representable resistances by allowing bridges, Plot2 of a(n)/A048211.
Crossrefs
Extensions
a(8) corrected and a(9)-a(17) from Rainer Rosenthal, Oct 29 2020
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