A338781 - OEIS

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

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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}