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A176501 a(n) = Farey(m; I) where m = Fibonacci (n + 1) and I = [1/n, 1]. 12
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 (list; graph; refs; listen; history; text; internal format)
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]

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

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-Fran├žois 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 and A174286, A176499, A176500, A176502.

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

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)