%I #50 Oct 21 2021 06:32:32

%S 0,1,3,7,15,35,79,193,493,1299,3429,9049,23699,62271,163997,433433,

%T 1147659,3040899

%N Number of distinct finite resistances that can be produced using at most n equal resistors (n or fewer resistors) in series, parallel and/or bridge configurations.

%C This sequence is a variation on A153588, which uses only series and parallel combinations. The circuits with exactly n unit resistors are counted by A174283, so this sequence counts the union of the sets, which are counted by A174283(k), k <= n. - _Rainer Rosenthal_, Oct 27 2020

%C For n = 0 the resistance is infinite, therefore the number of finite resistances is a(0) = 0. Sequence A180414 counts all resistances (including infinity) and so has A180414(0) = 1 and A180414(n) = a(n) + 1 for all n up to n = 7. For n > 7 the networks get more complex, producing more resistance values, so A180414(n) > a(n) + 1. - _Rainer Rosenthal_, Feb 13 2021

%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 Rainer Rosenthal, <a href="/A174284/a174284.txt">Maple program SetA174283 used for A174284</a>

%H <a href="/index/Res#resistances">Index to sequences related to resistances</a>.

%F a(n) = #(union of all S(k), k <= n), where S(k) is the set which is counted by A174283(k). - _Rainer Rosenthal_, Oct 27 2020

%e Since a bridge circuit requires a minimum of five resistors, the first four terms coincide with A153588. The fifth term also coincides since the set corresponding to five resistors for the bridge, i.e. {1}, is already obtained in the fourth set corresponding to the fourth term in A153588. [Edited by _Rainer Rosenthal_, Oct 27 2020]

%p # SetA174283(n) is the set of resistances counted by A174283(n) (see Maple link).

%p AccumulatedSetsA174283 := proc(n) option remember;

%p if n=1 then {1} else `union`(AccumulatedSetsA174283(n-1), SetA174283(n)) fi end:

%p A174284 := n -> nops(AccumulatedSetsA174283(n)):

%p seq(A174284(n), n=1..9); # _Rainer Rosenthal_, Oct 27 2020

%Y Cf. A048211, A153588, A174283, A174285, A174286, A176499, A176500, A176501, A176502, A180414.

%K more,nonn

%O 0,3

%A _Sameen Ahmed Khan_, Mar 15 2010

%E a(8) corrected, a(9)-a(17) from _Rainer Rosenthal_, Oct 27 2020

%E Title changed and a(0) added by _Rainer Rosenthal_, Feb 13 2021