cp's OEIS Frontend

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.

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