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A051389 Number of rational resistances requiring exactly n 1-ohm resistors in series or parallel. 7
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 (list; graph; refs; listen; history; text; internal format)
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.

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.

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

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Last modified August 4 02:24 EDT 2021. Contains 346441 sequences. (Running on oeis4.)