A051389 Number of rational resistances requiring exactly n 1-ohm resistors in series or parallel.

1, 2, 4, 8, 20, 42, 102, 250, 610, 1486, 3710, 9228, 23050, 57718, 145288, 365820, 922194, 2327914, 5885800, 14890796, 37701452, 95550472, 242325118, 614869792, 1561228066

Offset

1,2

Comments

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).

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).

Links

Table of n, a(n) for n=1..25.

Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.

Index to sequences related to resistances.

Formula

a(n) = A153588(n) - A153588(n-1) for n > 1. - Hugo Pfoertner, Nov 04 2020

Example

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).

Crossrefs

Cf. A048211, A046825, A153588, A180414, A338197.

Sequence in context: A272122 A323019 A105319 * A078006 A288476 A338197

Adjacent sequences: A051386 A051387 A051388 * A051390 A051391 A051392

Keyword

nonn,nice,more

Author

Hugo van der Sanden

Extensions

a(15)-a(21) from Jon E. Schoenfield, Aug 28 2006

Definition corrected by Jon E. Schoenfield, Aug 27 2006

a(22)-a(23) from Graeme McRae, Aug 18 2007

a(24)-a(25) from Antoine Mathys, Mar 20 2017

Definition changed to say "exactly". - N. J. A. Sloane, Nov 07 2020

Status

approved