%I #26 Feb 25 2023 15:44:52
%S 1,3,10,38,161,718,3385,16548,83183,427490,2237196,11865560,63677761
%N 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.
%C In order to get the maximum number, the ratio of the two resistances should be a transcendental number.
%C 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.
%H David Einstein, <a href="/A338781/a338781.pdf">Proof that the maximum number of resistances is even</a>.
%e In the following let x and y be the values of the two resistors.
%e With 1 component the resistances are {x, y}, so a(1) = 2/2 = 1.
%e 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.
%o (PARI)
%o 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])))))}
%o 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}
%o a(n)={#(S(n)[n])/2}
%Y Cf. A048211.
%K nonn,more
%O 1,2
%A _Andrew Howroyd_, Nov 08 2020
%E a(11) from _Alois P. Heinz_, Dec 21 2020
%E a(12)-a(13) from _David Einstein_, Feb 23 2022