A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).

2, 3, 5, 11, 23, 59, 141, 361, 941, 2457, 6331, 16619, 43359, 113159, 296385, 775897, 2030103, 5315385, 13912615, 36421835, 95355147, 249635525, 653525857, 1710966825, 4479358275, 11726974249, 30701593527, 80377757397, 210431301141, 550916379293

Offset: 1

Author

Sameen Ahmed Khan, Apr 21 2010

Keywords

• nonn

Comments

This sequence arises in the analytically obtained 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 a strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176501 provides a better strict upper bound but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

Farey(n) = A005728(n). [Franklin T. Adams-Watters, May 12 2010]

The claim that this sequence is a strict upper bound for the number of representable resistance values of any conceivable network is wrong. It only applies to purely serial-parallel networks (A048211), but it already fails when bridges are allowed, as described in A174283. Even more so if arbitrary nonplanar networks are allowed as in A337517. See the linked illustrations of the respective quotients. - Hugo Pfoertner, Jan 24 2021

Examples

			n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23.
		

Links

• Antoine Mathys, Table of n, a(n) for n = 1..50
• 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). Digital Object Identifier (DOI): 10.1119/1.19396.
• 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, Mathematica notebook 1
• Sameen Ahmed KHAN, Mathematica notebook 2
• Hugo Pfoertner, Ratio for series-parallel networks, Plot2 of A048211(n)/a(n).
• Hugo Pfoertner, Ratio for networks with bridges, Plot2 of A174283(n)/a(n).
• Hugo Pfoertner, Ratio for arbitrary networks, Plot2 of A337517(n)/a(n).

Crossrefs

Cf. A000045, A005728.

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176500, A176501, A176502, A337517.

Programs

GAP

List([1..30],n->Sum([1..Fibonacci(n+1)],i->Phi(i)))+1; # Muniru A Asiru, Jul 31 2018

Magma

[1+&+[EulerPhi(i):i in [1..Fibonacci(n+1)]]:n in [1..30]]; // Marius A. Burtea, Jul 26 2019

Maple

with(numtheory): with(combinat,fibonacci): a:=n->1+add(phi(i),i=1..n): seq(a(fibonacci(n+1)),n=1..30); # Muniru A Asiru, Jul 31 2018

Mathematica

b[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
a[n_] := b[Fibonacci[n + 1]];
Array[a, 30] (* Jean-François Alcover, Sep 20 2018 *)

PARI

farey(n) = 1+sum(k=1, n, eulerphi(k));
a(n) = farey(fibonacci(n+1)); \\ Michel Marcus, Jul 31 2018

Formula

a(n) = A005728(A000045(n+1)). - Michel Marcus, Jul 31 2018

Extensions

a(26)-a(29) from Sameen Ahmed Khan, May 02 2010

a(30) from Antoine Mathys, Aug 06 2018