A048211 Number of distinct resistances that can be produced from a circuit of n equal resistors using only series and parallel combinations.

1, 2, 4, 9, 22, 53, 131, 337, 869, 2213, 5691, 14517, 37017, 93731, 237465, 601093, 1519815, 3842575, 9720769, 24599577, 62283535, 157807915, 400094029, 1014905643, 2576046289, 6541989261, 16621908599, 42251728111, 107445714789, 273335703079

Offset: 1

Author

Tony Bartoletti

Keywords

• nonn
• nice
• more
• hard

Comments

Found by exhaustive search. Program produces all values that are combinations of two binary operators a() and b() (here "sum" and "reciprocal sum of reciprocals") over n occurrences of 1. E.g., given 4 occurrences of 1, the code forms all allowable postfix forms, such as 1 1 1 1 a a a and 1 1 b 1 1 a b, etc. Each resulting form is then evaluated according to the definitions for a and b.

Each resistance that can be constructed from n 1-ohm resistors in a circuit can be written as the ratio of two positive integers, neither of which exceeds the (n+1)st Fibonacci number. E.g., for n=4, the 9 resistances that can be constructed can be written as 1/4, 2/5, 3/5, 3/4, 1/1, 4/3, 5/3, 5/2, 4/1 using no numerator or denominator larger than Fib(n+1) = Fib(5) = 5. If a resistance x can be constructed from n 1-ohm resistors, then a resistance 1/x can also be constructed from n 1-ohm resistors. - Jon E. Schoenfield, Aug 06 2006

The fractions in the comment above are a superset of the fractions occurring here, corresponding to the upper bound A176500. - Joerg Arndt, Mar 07 2015

The terms of this sequence consider only series and parallel combinations; A174283 considers bridge combinations as well. - Jon E. Schoenfield, Sep 02 2013

Examples

			a(2) = 2 since given two 1-ohm resistors, a series circuit yields 2 ohms, while a parallel circuit yields 1/2 ohms.
		

Links

• Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000). [From _Sameen Ahmed Khan_, Apr 27 2010]
• Sameen Ahmed Khan, Mathematica program
• Sameen Ahmed Khan, Mathematica notebook for A048211 and A000084
• Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346 [physics.gen-ph], 2010.
• Sameen Ahmed Khan, How Many Equivalent Resistances?, RESONANCE, May 2012. - From _N. J. A. Sloane_, Oct 15 2012
• Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From _N. J. A. Sloane_, Oct 23 2012
• Sameen Ahmed Khan, Beginning to Count the Number of Equivalent Resistances, Indian Journal of Science and Technology, Vol. 9, Issue 44, pp. 1-7, 2016.
• Marx Stampfli, Bridged graphs, circuits and Fibonacci numbers, Applied Mathematics and Computation, Volume 302, 1 June 2017, Pages 68-79.

Crossrefs

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 A051389 is h(n) = sum(H(x, n)) for all x (x is necessarily rational).

Cf. A153588, A174283, A174284, A174285 and A174286, A176497, A176498, A176499, A176500, A176501, A176502. - Sameen Ahmed Khan, Apr 27 2010

Cf. A180414.

Programs

Maple

r:= proc(n) option remember; `if`(n=1, {1}, {seq(seq(seq(
      [f+g, 1/(1/f+1/g)][], g in r(n-i)), f in r(i)), i=1..n/2)})
    end:
a:= n-> nops(r(n)):
seq(a(n), n=1..15);  # Alois P. Heinz, Apr 02 2015

Mathematica

r[n_] := r[n] = If[n == 1, {1}, Union @ Flatten @ {Table[ Table[ Table[ {f+g, 1/(1/f+1/g)}, {g, r[n-i]}], {f, r[i]}], {i, 1, n/2}]}]; a[n_] := Length[r[n]]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

PARI

\\ not efficient; just to show the method
N=10;
L=vector(N);  L[1]=[1];
{ for (n=2, N,
    my( T = Set( [] ) );
    for (k=1, n\2,
        for (j=1, #L[k],
            my( r1 = L[k][j] );
            for (i=1, #L[n-k],
                my( r2 = L[n-k][i] );
                T = setunion(T,  Set([r1+r2, r1*r2/(r1+r2) ]) );
            );
        );
    );
    T = vecsort(Vec(T), , 8);
    L[n] = T;
); }
for(n=1, N, print1(#L[n], ", ") );
\\ Joerg Arndt, Mar 07 2015

Formula

From Bill McEachen, Jun 08 2024: (Start)

(2.414^n)/4 < a(n) < (1-1/n)*(0.318)*(2.618^n) (Khan, n>3).

Conjecture: a(n) ~ K * a(n-1), K approx 2.54. (End)

Extensions

More terms from John W. Layman, Apr 06 2002

a(16)-a(21) from Jon E. Schoenfield, Aug 06 2006

a(22) from Jon E. Schoenfield, Aug 28 2006

a(23) from Jon E. Schoenfield, Apr 18 2010

Definition edited (to specify that the sequence considers only series and parallel combinations) by Jon E. Schoenfield, Sep 02 2013

a(24)-a(25) from Antoine Mathys, Apr 02 2015

a(26)-a(27) from Johannes P. Reichart, Nov 24 2018

a(28)-a(30) from Antoine Mathys, Dec 08 2024