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A176500 a(n) = 2*Farey(Fibonacci(n + 1)) - 3. 13

%I

%S 1,3,7,19,43,115,279,719,1879,4911,12659,33235,86715,226315,592767,

%T 1551791,4060203,10630767,27825227,72843667,190710291,499271047,

%U 1307051711,3421933647,8958716547,23453948495,61403187051,160755514791,420862602279,1101832758583

%N a(n) = 2*Farey(Fibonacci(n + 1)) - 3.

%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. A176502 provides a better strict upper bound but is harder to compute. [Corrected by _Antoine Mathys_, Jul 12 2019]

%C Farey(n) = A005728(n). - _Franklin T. Adams-Watters_, May 12 2010

%H Antoine Mathys, <a href="/A176500/b176500.txt">Table of n, a(n) for n = 1..50</a>

%H Antoni Amengual, <a href="https://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], (Apr 20 2010).

%H Sameen Ahmed KHAN, <a href="/A176500/a176500a.nb">Mathematica notebook 1</a>

%H Sameen Ahmed KHAN, <a href="/A176500/a176500b.nb">Mathematica notebook 2</a>

%F a(n) = 2 * A176499(n) - 3.

%e n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23, 2Farey(m) - 3 = 43.

%t a[n_] := 2 Sum[EulerPhi[k], {k, 1, Fibonacci[n+1]}] - 1;

%t Table[an = a[n]; Print[an]; an, {n, 1, 30}] (* _Jean-Fran├žois Alcover_, Nov 03 2018, from PARI *)

%o (PARI) a(n) = 2*sum(k=1,fibonacci(n+1),eulerphi(k))-1 \\ _Charles R Greathouse IV_, Oct 07 2016

%o (MAGMA) [2*(&+[EulerPhi(k):k in [1..Fibonacci(n+1)]])-1:n in [1..30]]; // _Marius A. Burtea_, Jul 26 2019

%Y Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176501, A176502.

%K nonn

%O 1,2

%A _Sameen Ahmed Khan_, Apr 21 2010

%E a(26)-a(28) from _Sameen Ahmed Khan_, May 02 2010

%E a(29)-a(30) from _Antoine Mathys_, Aug 06 2018

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