cp's OEIS Frontend

A337516 Number of distinct resistances that can be produced using n unit resistors in series, parallel, bridge or fork configurations.

%I A337516 #27 Sep 02 2023 15:37:14
%S A337516 1,2,4,9,23,57,151,421,1202,3397,9498,25970,70005,187259,500061
%N A337516 Number of distinct resistances that can be produced using n unit resistors in series, parallel, bridge or fork configurations.
%C A337516 Each network with 2, 3 or 4 resistors is made up of series or parallel connected resistors in such a way that the resulting resistances can be computed as Ser(x1,x2) = x1 + x2 (type S) or Par(x1,x2) = 1/(1/x1+1/x2) (type P). The parameters are either 1 Ohm or themselves of type S or P. A048211 counts the different resistances which can be produced as S or P type from n unit resistors. With 5 resistors x1 .. x5 there is the bridge configuration (type B),
%C A337516         A         which cannot be computed by functions Ser() and Par().
%C A337516        / \        The resistance between A and D is given by
%C A337516      x1   x2
%C A337516      /     \      Bri(x1,x2,x3,x4,x5) =
%C A337516     B- x3 - C
%C A337516      \     /      x2*x1*x4+x2*x1*x5+x5*x4*x1+x5*x4*x2+x3*(x2+x5)*(x1+x4)
%C A337516      x4   x5      ------------------------------------------------------ .
%C A337516        \ /                  (x1+x2)*(x4+x5)+x3*(x1+x4+x2+x5)
%C A337516         D
%C A337516 Sequence A174283 counts all resistances of types S, P and B which can be produced with n unit resistors. The next essentially new figuration comes with 7 resistors: the fork (type F), which cannot be computed by functions Ser(), Par() and Bri().
%C A337516               A
%C A337516              / \
%C A337516            x3   x1
%C A337516            /     \
%C A337516           B- x5 - C
%C A337516          / \     /
%C A337516        x4  x7   x6
%C A337516        /     \ /
%C A337516       E- x2 - D
%C A337516 The resistance between A and E is given by
%C A337516 Frk(x1,x2,x3,x4,x5,x6,x7) =
%C A337516 x1*x3*x4*x7+x1*x3*x4*x5+x1*x3*x2*x7+x1*x3*x2*x5+x2*x4*x3*x7+x2*x4*x3*x5+
%C A337516 x2*x4*x1*x7+x2*x4*x1*x5+x5*x7*x1*x3+x5*x7*x1*x4+x5*x7*x2*x3+x5*x7*x2*x4+
%C A337516 x6*x1*x3*x7+x6*x1*x3*x2+x6*x1*x3*x4+x6*x5*x7*x3+x6*x5*x2*x3+x6*x3*x4*x5+
%C A337516 x6*x3*x4*x7+x6*x1*x4*x7+x6*x5*x7*x4+x6*x2*x4*x3+x6*x2*x4*x1+x6*x5*x2*x4
%C A337516 ------------------------------------------------------------------------ .
%C A337516    x3*x4*x7+x3*x4*x5+x2*x3*x7+x5*x2*x3+x1*x4*x7+x5*x1*x4+x1*x2*x7+
%C A337516    x1*x2*x5+x5*x7*x3+x5*x7*x4+x5*x7*x1+x5*x7*x2+x6*x3*x7+x6*x2*x3+
%C A337516    x6*x3*x4+x6*x1*x7+x6*x1*x2+x6*x1*x4+x6*x5*x7+x6*x5*x2+x6*x4*x5
%C A337516 This sequence A337516 counts all resistances of type S, P, B or F which can be produced with n unit resistors.
%H A337516 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A337516&amp;name2=A174283&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true&amp;drawlines=true">Increase in the number of representable resistance values through the fork bridge type</a>, Plot2 of a(n)/A174283(n).
%H A337516 Rainer Rosenthal, <a href="/A337516/a337516.txt">Maple program SetA337516 to generate the sets counted by A337516</a>
%e A337516 a(1) through a(6) are identical with A174283 since a fork needs at least 7 resistors. a(7) is also equal to A174283(7) because the fork with 7 unit resistors has resistance 8/7, but this is already an element of SetA174283(7).
%e A337516 a(8) = 421 has six extra resistances {16/17, 40/29, 35/34, 37/29, 35/31, 37/32} which are the result of resistance 2 or 1/2 as any of the resistances x1 .. x7 except for x6.
%p A337516 # SetA337516(n) is the set of resistances counted by A337516(n) (see Maple link).
%p A337516 A337516 := n -> nops(SetA337516(n)):
%p A337516 seq(A337516(n), n=1..9);
%Y A337516 Cf. A048211, A174283, A337517.
%K A337516 nonn,hard,nice,more
%O A337516 1,2
%A A337516 _Rainer Rosenthal_, Oct 29 2020