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%I #47 Jan 23 2021 17:33:19
%S 1,2,4,9,23,57,151,415,1157,3191,8687,23199,61677,163257,432541,
%T 1146671,3039829
%N Number of distinct resistances that can be produced using n equal resistors in, series, parallel and/or bridge configurations.
%C 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.
%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2), 175-179 (February 2000).
%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346/">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).
%H Sameen Ahmed Khan, <a href="https://www.ias.ac.in/describe/article/pmsc/122/02/0153-0162">Farey sequences and resistor networks</a>, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162.
%H Sameen Ahmed Khan, <a href="https://dx.doi.org/10.17485/ijst/2016/v9i44/88086">Beginning to Count the Number of Equivalent Resistances</a>, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 1-7, 2016.
%H Hugo Pfoertner, <a href="/plot2a?name1=A174283&name2=A048211&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true&drawlines=true">Increase of number of representable resistances by allowing bridges</a>, Plot2 of a(n)/A048211.
%e 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.
%e 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
%Y Cf. A048211, A153588, A174284, A174285, A174286, A176499, A176500, A176501, A176502, A180414, A337516, A337517.
%K nonn,hard,nice,more
%O 1,2
%A _Sameen Ahmed Khan_, Mar 15 2010
%E a(8) corrected and a(9)-a(17) from _Rainer Rosenthal_, Oct 29 2020
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