%I #40 May 08 2023 15:22:08
%S 1,2,4,8,20,42,102,250,610,1486,3710,9228,23050,57718,145288,365820,
%T 922194,2327914,5885800,14890796,37701452,95550472,242325118,
%U 614869792,1561228066
%N Number of rational resistances requiring exactly n 1-ohm resistors in series or parallel.
%C If x and y require xn and yn resistors respectively, then (x+y) and 1/(1/x + 1/y) require no more than (xn+yn). Inspired by a sci.math posting by Miguel A. Lerma (lerma(AT)math.nwu.edu).
%C Let T(x, n) = 1 if x can be constructed with n 1-ohm resistors in a circuit, 0 otherwise. Then A048211 is t(n) = sum(T(x, n)) for all x (x is necessarily rational). Let H(x, n) = 1 if T(x, n) = 1 and T(x, k) = 0 for all k < n, 0 otherwise. Then a(n) is h(n) = sum(H(x, n)) for all x (x is necessarily rational).
%H Miguel A. Lerma, <a href="https://groups.google.com/g/sci.math/c/6OUCm6hxJpM/m/M2b1UBeGH60J">resistors</a>, post in the newsgroup sci.math, Nov 5 1999.
%H <a href="/index/Res#resistances">Index to sequences related to resistances</a>.
%F a(n) = A153588(n) - A153588(n-1) for n > 1. - _Hugo Pfoertner_, Nov 04 2020
%e a(5)=card({5,1/5,5/4,4/5,7/3,3/7,7/4,4/7,7/2,2/7,7/5,5/7,8/3,3/8,8/5,5/8, 5/6,7/6,6/5,7/6}). E.g. 6/5 is made from two resistors in series in parallel with three resistors in series, since 6/5 = 1/(1/2 + 1/3).
%Y Cf. A048211, A046825, A153588, A180414, A338197.
%K nonn,nice,more
%O 1,2
%A _Hugo van der Sanden_
%E a(15)-a(21) from _Jon E. Schoenfield_, Aug 28 2006
%E Definition corrected by _Jon E. Schoenfield_, Aug 27 2006
%E a(22)-a(23) from _Graeme McRae_, Aug 18 2007
%E a(24)-a(25) from _Antoine Mathys_, Mar 20 2017
%E Definition changed to say "exactly". - _N. J. A. Sloane_, Nov 07 2020