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

1,2

Comments

This sequence is a variation on A048211, which uses only series and parallel combinations. Since a bridge circuit requires minimum of five resistances the first four terms coincide. For the definition of "bridge" see A337516.

Links

Table of n, a(n) for n=1..17.

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.

Example

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

Crossrefs

Cf. A048211, A153588, A174284, A174285, A174286, A176499, A176500, A176501, A176502, A180414, A337516, A337517.

Sequence in context: A159330 A159331 A135346 * A337516 A340920 A337517

Adjacent sequences: A174280 A174281 A174282 * A174284 A174285 A174286

Keyword

nonn,hard,nice,more

Author

Sameen Ahmed Khan, Mar 15 2010

Extensions

a(8) corrected and a(9)-a(17) from Rainer Rosenthal, Oct 29 2020

Status

approved