A176501 a(n) = Farey(m; I) where m = Fibonacci (n + 1) and I = [1/n, 1].

1, 2, 4, 9, 19, 50, 122, 317, 837, 2213, 5758, 15236, 40028, 105079, 276627, 727409, 1910685, 5020094, 13180380, 34600740, 90814431, 238288480, 625111687, 1639676484, 4300183922, 11275936787, 29564497466, 77507123132, 203175049457, 532552499826, 1395790412496

Offset

1,2

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 an strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. This sequence provides a better strict upper bound than A176499 but is harder to compute. [Corrected by Antoine Mathys, May 07 2019]

From Hugo Pfoertner, Jan 24 2021: (Start)

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, 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.

But in contrast to A176499, which at least correctly bounds A048211, the terms a(5), ..., a(9) in this sequence are smaller than the corresponding terms from A048211 (a(n) vs. A048211(n): 19/22, 50/53, 122/131, 317/337, 837/869). (End)

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, Mathematica notebook

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).

Example

n = 5, I = [1/5, 1], m = Fibonacci(5 + 1) = 8, Farey(8) = 23, Farey(8; I) = 19

Mathematica

a[n_ /; n<4] := 2^(n-1); a[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]];
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;
a(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; \\ Antoine Mathys, May 07 2019

Crossrefs

Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176502, A337517.

Sequence in context: A327017 A153447 A076893 * A324317 A096354 A076838

Adjacent sequences: A176498 A176499 A176500 * A176502 A176503 A176504

Keyword

nonn

Author

Sameen Ahmed Khan, Apr 21 2010

Extensions

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

a(28)-a(31) from Antoine Mathys, May 07 2019

Status

approved