A176502 a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1].

1, 3, 7, 17, 37, 99, 243, 633, 1673, 4425, 11515, 30471, 80055, 210157, 553253, 1454817, 3821369, 10040187, 26360759, 69201479, 181628861, 476576959, 1250223373, 3279352967, 8600367843, 22551873573, 59128994931, 155014246263, 406350098913, 1065104999651

Offset

1,2

Comments

This sequence provides a strict upper bound of the set of equivalent resistances formed by any conceivable network (series/parallel or bridge, or non-planar) of n equal resistors. Consequently it provides an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176500 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is incorrect for networks with more than 10 resistors, in which non-planar configurations can also occur. Whether the sequence provides at least a valid upper bound for planar networks with generalized bridge circuits (A337516) is difficult to decide on the basis of the insufficient number of terms in A174283 and A337516. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 25 2021

Links

Antoine Mathys, Table of n, a(n) for n = 1..40

Antoni Amengual, The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel, American Journal of Physics, 68(2), 175-179 (February 2000).

Sameen Ahmed Khan, The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).

Sameen Ahmed Khan, Integer Sequences Authored by Dr. Sameen Ahmed Khan

Sameen Ahmed Khan, Mathematica notebook

Sameen Ahmed Khan, How Many Equivalent Resistances?, RESONANCE, May 2012. - From N. J. A. Sloane, Oct 15 2012

Sameen Ahmed Khan, Farey sequences and resistor networks, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From N. J. A. Sloane, Oct 23 2012

Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).

Hugo Pfoertner, Ratio for planar networks with generalized bridges, Plot2 of A337516(n)/a(n).

Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Formula

a(n) = 2 * A176501(n) - 1. - Antoine Mathys, Aug 07 2018

Example

n = 5, , I = [1/5, 1], m = Fibonacci(6) = 8, Farey(8) = 23, Farey(8; I) = 19, Grand Set(5) = 37.

Mathematica

a1[n_ /; n<4] := 2^(n-1); a1[n_] := Module[{m = Fibonacci[n+1], v}, v = Reap[Do[Sow[j/i], {i, n+1, m}, {j, 1, (i-1)/n}]][[2, 1]]; Total[EulerPhi[ Range[m]]] - Length[v // Union]];
a[n_] := 2 a1[n] - 1;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* Jean-François Alcover, Aug 30 2018, after Antoine Mathys *)

Prog

(PARI) farey(n) = sum(i=1, n, eulerphi(i)) + 1;
a176501(n) = my(m=fibonacci(n + 1), count=0); for(b=n+1, m, for(a=1, (b-1)/n, if(gcd(a, b)==1, count++))); farey(m) - 1 - count;
a(n) = 2 * a176501(n) - 1; \\ Antoine Mathys, May 07 2019

Crossrefs

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176501, A337516, A337517.

Sequence in context: A178155 A330457 A026396 * A319003 A141199 A368655

Adjacent sequences: A176499 A176500 A176501 * A176503 A176504 A176505

Keyword

nonn

Author

Sameen Ahmed Khan, Apr 21 2010

Extensions

a(19)-a(27) from Antoine Mathys, Aug 10 2018

a(28)-a(30) from Antoine Mathys, May 07 2019

Status

approved