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

Author

Andrew Howroyd, Nov 08 2020

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A048211.

Programs

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}

Extensions

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

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