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

1, 2, 4, 9, 23, 57, 151, 421, 1202, 3397, 9498, 25970, 70005, 187259, 500061

Offset: 1

Author

Rainer Rosenthal, Oct 29 2020

Keywords

• nonn
• hard
• nice
• more

Comments

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),

A which cannot be computed by functions Ser() and Par().

/ \ The resistance between A and D is given by

x1 x2

/ \ Bri(x1,x2,x3,x4,x5) =

B- x3 - C

\ / x2*x1*x4+x2*x1*x5+x5*x4*x1+x5*x4*x2+x3*(x2+x5)*(x1+x4)

x4 x5 ------------------------------------------------------ .

\ / (x1+x2)*(x4+x5)+x3*(x1+x4+x2+x5)

D

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().

A

/ \

x3 x1

/ \

B- x5 - C

/ \ /

x4 x7 x6

/ \ /

E- x2 - D

The resistance between A and E is given by

Frk(x1,x2,x3,x4,x5,x6,x7) =

x1*x3*x4*x7+x1*x3*x4*x5+x1*x3*x2*x7+x1*x3*x2*x5+x2*x4*x3*x7+x2*x4*x3*x5+

x2*x4*x1*x7+x2*x4*x1*x5+x5*x7*x1*x3+x5*x7*x1*x4+x5*x7*x2*x3+x5*x7*x2*x4+

x6*x1*x3*x7+x6*x1*x3*x2+x6*x1*x3*x4+x6*x5*x7*x3+x6*x5*x2*x3+x6*x3*x4*x5+

x6*x3*x4*x7+x6*x1*x4*x7+x6*x5*x7*x4+x6*x2*x4*x3+x6*x2*x4*x1+x6*x5*x2*x4

------------------------------------------------------------------------ .

x3*x4*x7+x3*x4*x5+x2*x3*x7+x5*x2*x3+x1*x4*x7+x5*x1*x4+x1*x2*x7+

x1*x2*x5+x5*x7*x3+x5*x7*x4+x5*x7*x1+x5*x7*x2+x6*x3*x7+x6*x2*x3+

x6*x3*x4+x6*x1*x7+x6*x1*x2+x6*x1*x4+x6*x5*x7+x6*x5*x2+x6*x4*x5

This sequence A337516 counts all resistances of type S, P, B or F which can be produced with n unit resistors.

Examples

			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).
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.
		

Links

• Hugo Pfoertner, Increase in the number of representable resistance values through the fork bridge type, Plot2 of a(n)/A174283(n).
• Rainer Rosenthal, Maple program SetA337516 to generate the sets counted by A337516

Crossrefs

Cf. A048211, A174283, A337517.

Programs

Maple

# SetA337516(n) is the set of resistances counted by A337516(n) (see Maple link).
A337516 := n -> nops(SetA337516(n)):
seq(A337516(n), n=1..9);