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%I #27 Sep 02 2023 15:37:14
%S 1,2,4,9,23,57,151,421,1202,3397,9498,25970,70005,187259,500061
%N Number of distinct resistances that can be produced using n unit resistors in series, parallel, bridge or fork configurations.
%C 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 A which cannot be computed by functions Ser() and Par().
%C / \ The resistance between A and D is given by
%C x1 x2
%C / \ Bri(x1,x2,x3,x4,x5) =
%C B- x3 - C
%C \ / x2*x1*x4+x2*x1*x5+x5*x4*x1+x5*x4*x2+x3*(x2+x5)*(x1+x4)
%C x4 x5 ------------------------------------------------------ .
%C \ / (x1+x2)*(x4+x5)+x3*(x1+x4+x2+x5)
%C D
%C 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 A
%C / \
%C x3 x1
%C / \
%C B- x5 - C
%C / \ /
%C x4 x7 x6
%C / \ /
%C E- x2 - D
%C The resistance between A and E is given by
%C Frk(x1,x2,x3,x4,x5,x6,x7) =
%C 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 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 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 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 ------------------------------------------------------------------------ .
%C x3*x4*x7+x3*x4*x5+x2*x3*x7+x5*x2*x3+x1*x4*x7+x5*x1*x4+x1*x2*x7+
%C x1*x2*x5+x5*x7*x3+x5*x7*x4+x5*x7*x1+x5*x7*x2+x6*x3*x7+x6*x2*x3+
%C x6*x3*x4+x6*x1*x7+x6*x1*x2+x6*x1*x4+x6*x5*x7+x6*x5*x2+x6*x4*x5
%C This sequence A337516 counts all resistances of type S, P, B or F which can be produced with n unit resistors.
%H Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A337516&name2=A174283&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true&drawlines=true">Increase in the number of representable resistance values through the fork bridge type</a>, Plot2 of a(n)/A174283(n).
%H Rainer Rosenthal, <a href="/A337516/a337516.txt">Maple program SetA337516 to generate the sets counted by A337516</a>
%e 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 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 # SetA337516(n) is the set of resistances counted by A337516(n) (see Maple link).
%p A337516 := n -> nops(SetA337516(n)):
%p seq(A337516(n), n=1..9);
%Y Cf. A048211, A174283, A337517.
%K nonn,hard,nice,more
%O 1,2
%A _Rainer Rosenthal_, Oct 29 2020
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