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A176502 a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1]. 14
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 (list; graph; refs; listen; history; text; internal format)
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]

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

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

S. A. 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

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-Fran├žois 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 and A174286, A176499, A176500, A176501.

Sequence in context: A178941 A178155 A026396 * A319003 A141199 A003478

Adjacent sequences:  A176499 A176500 A176501 * A176503 A176504 A176505

KEYWORD

more,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

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Last modified October 14 14:45 EDT 2019. Contains 328019 sequences. (Running on oeis4.)