A338781 Half the maximum number of distinct resistances that can be produced from a circuit of n resistors of two different kinds using only series and parallel combinations.

1, 3, 10, 38, 161, 718, 3385, 16548, 83183, 427490, 2237196, 11865560, 63677761

Offset

1,2

Comments

In order to get the maximum number, the ratio of the two resistances should be a transcendental number.

It appears that the resistance values always come in pairs, but this has not been proven. (This sequence only enumerates half). In particular, it seems that switching the two types of resistor and exchanging parallel with serial will always give a different value. Neither of these on its own is sufficient.

Links

Table of n, a(n) for n=1..13.

David Einstein, Proof that the maximum number of resistances is even.

Example

In the following let x and y be the values of the two resistors.
With 1 component the resistances are {x, y}, so a(1) = 2/2 = 1.
With 2 components the resistances are {2*x, x/2, 2*y, y/2, x + y, x*y/(x + y)}, so a(2) = 6/2 = 3.

Prog

(PARI)
ParSer(u, v)={concat(concat(vector(#u, i, vector(#v, j, u[i]+v[j]))), concat(vector(#u, i, vector(#v, j, 1/(1/u[i]+1/v[j])))))}
S(n)={my(v=vector(n)); v[1]=[1, 'x]; for(n=2, #v, v[n]=Set(concat(vector(n\2, k, ParSer(v[k], v[n-k]))))); v}
a(n)={#(S(n)[n])/2}

Crossrefs

Cf. A048211.

Sequence in context: A151063 A103138 A074527 * A359109 A346815 A351681

Adjacent sequences: A338778 A338779 A338780 * A338782 A338783 A338784

Keyword

nonn,more

Author

Andrew Howroyd, Nov 08 2020

Extensions

a(11) from Alois P. Heinz, Dec 21 2020

a(12)-a(13) from David Einstein, Feb 23 2022

Status

approved