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 A176502 a(n) = 2*Farey(m; I) - 1 where m = Fibonacci (n + 1) and I = [1/n, 1]. 14

%I

%S 1,3,7,17,37,99,243,633,1673,4425,11515,30471,80055,210157,553253,

%T 1454817,3821369,10040187,26360759,69201479,181628861,476576959,

%U 1250223373,3279352967,8600367843,22551873573,59128994931,155014246263,406350098913,1065104999651

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

%C 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]

%H Antoine Mathys, <a href="/A176502/b176502.txt">Table of n, a(n) for n = 1..40</a>

%H Antoni Amengual, <a href="http://dx.doi.org/10.1119/1.19396">The intriguing properties of the equivalent resistances of n equal resistors combined in series and in parallel</a>, American Journal of Physics, 68(2), 175-179 (February 2000).

%H Sameen Ahmed Khan, <a href="http://arxiv.org/abs/1004.3346/">The bounds of the set of equivalent resistances of n equal resistors combined in series and in parallel</a>, arXiv:1004.3346v1 [physics.gen-ph], (20 April 2010).

%H Sameen Ahmed Khan, <a href="http://sameenahmedkhan.webs.com/integer-sequences.html">Integer Sequences Authored by Dr. Sameen Ahmed Khan</a>

%H Sameen Ahmed Khan, <a href="/A176502/a176502.nb">Mathematica notebook</a>

%H S. A. Khan, <a href="http://www.ias.ac.in/resonance/May2012/p468-475.pdf">How Many Equivalent Resistances?</a>, RESONANCE, May 2012. - From _N. J. A. Sloane_, Oct 15 2012

%H S. A. Khan, <a href="http://www.ias.ac.in/mathsci/vol122/may2012/pmsc-d-10-00141.pdf">Farey sequences and resistor networks</a>, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 122, No. 2, May 2012, pp. 153-162. - From _N. J. A. Sloane_, Oct 23 2012

%F a(n) = 2 * A176501(n) - 1. - _Antoine Mathys_, Aug 07 2018

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

%t 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]];

%t a[n_] := 2 a1[n] - 1;

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* _Jean-François Alcover_, Aug 30 2018, after _Antoine Mathys_ *)

%o (PARI) farey(n) = sum(i=1, n, eulerphi(i)) + 1;

%o 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;

%o a(n) = 2 * a176501(n) - 1; \\ _Antoine Mathys_, May 07 2019

%Y Cf. A048211, A153588, A174283, A174284, A174285 and A174286, A176499, A176500, A176501.

%K more,nonn

%O 1,2

%A _Sameen Ahmed Khan_, Apr 21 2010

%E a(19)-a(27) from _Antoine Mathys_, Aug 10 2018

%E a(28)-a(30) from _Antoine Mathys_, May 07 2019

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Last modified June 5 11:52 EDT 2020. Contains 334840 sequences. (Running on oeis4.)