A051389 Number of resistance values that can be constructed using exactly n 1-ohm resistors in series or parallel but not with fewer resistors.

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, 3966071764, 10080113232, 25630109268, 65194419268, 165890640468

Offset: 1

Author

Hugo van der Sanden

Keywords

• nonn
• nice
• more

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 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

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

Examples

			The a(1) = 1 resistance value is 1 ohm.
The a(2) = 2 resistance values are {1/2, 2}.
The a(3) = 4 resistance values are {1/3, 2/3, 3/2, 3}.
The a(4) = 8 resistance values are {1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4}.
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.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.
		

Links

• Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.
• Index to sequences related to resistances.

Crossrefs

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

Formula

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

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

Definition clarified by Antoine Mathys, Nov 22 2024

a(26)-a(30) from Antoine Mathys, Dec 05 2024