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 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
LINKS
Miguel A. Lerma, resistors, post in the newsgroup sci.math, Nov 5 1999.
FORMULA
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,nice,more
AUTHOR
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
STATUS
approved