This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051389 #67 Dec 20 2024 22:32:17
%S A051389 1,2,4,8,20,42,102,250,610,1486,3710,9228,23050,57718,145288,365820,
%T A051389 922194,2327914,5885800,14890796,37701452,95550472,242325118,
%U A051389 614869792,1561228066,3966071764,10080113232,25630109268,65194419268,165890640468
%N A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.
%C A051389 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 A051389 Let A(n) be the set of resistances equivalent to a network of n 1-ohm resistors using only series and parallel combinations. Then A048211(n) = card(A(n)). Let L(n) be the set of resistances that first appear in A(n), i.e. L(n) = A(n) \ (A(1) U ... U A(n-1)). Then a(n) = card(L(n)). - _Antoine Mathys_, Nov 22 2024
%C A051389 If a resistance is equivalent to a n-resistor circuit, then it is equivalent to a 4n-resistor circuit. There is therefore no upper bound on the size of the networks to which it is equivalent. - _Antoine Mathys_, Nov 22 2024
%H A051389 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 A051389 <a href="/index/Res#resistances">Index to sequences related to resistances</a>.
%F A051389 a(n) = A153588(n) - A153588(n-1) for n > 1. - _Hugo Pfoertner_, Nov 04 2020
%e A051389 The a(1) = 1 resistance value is 1 ohm.
%e A051389 The a(2) = 2 resistance values are {1/2, 2}.
%e A051389 The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
%e A051389 The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
%e A051389 The a(5) = 20 resistance values are {1/5, 2/7, 3/8, 3/7, 4/7, 5/8, 5/7, 4/5, 5/6, 6/7, 7/6, 6/5, 5/4, 7/5, 8/5, 7/4, 7/3, 8/3, 7/2, 5}.
%e A051389 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). It cannot be obtained using fewer resistors.
%Y A051389 Cf. A048211, A046825, A153588, A180414, A338197.
%K A051389 nonn,nice,more
%O A051389 1,2
%A A051389 _Hugo van der Sanden_
%E A051389 a(15)-a(21) from _Jon E. Schoenfield_, Aug 28 2006
%E A051389 Definition corrected by _Jon E. Schoenfield_, Aug 27 2006
%E A051389 a(22)-a(23) from _Graeme McRae_, Aug 18 2007
%E A051389 a(24)-a(25) from _Antoine Mathys_, Mar 20 2017
%E A051389 Definition changed to say "exactly". - _N. J. A. Sloane_, Nov 07 2020
%E A051389 Definition clarified by _Antoine Mathys_, Nov 22 2024
%E A051389 a(26)-a(30) from _Antoine Mathys_, Dec 05 2024