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A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).

%I A176499 #54 Sep 08 2022 08:45:52
%S A176499 2,3,5,11,23,59,141,361,941,2457,6331,16619,43359,113159,296385,
%T A176499 775897,2030103,5315385,13912615,36421835,95355147,249635525,
%U A176499 653525857,1710966825,4479358275,11726974249,30701593527,80377757397,210431301141,550916379293
%N A176499 Haros-Farey sequence whose argument is the Fibonacci number; Farey(m) where m = Fibonacci(n + 1).
%C A176499 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 a strict upper bound of the sequences: A048211, A153588, A174283, A174284, A174285 and A174286. A176501 provides a better strict upper bound but is harder to compute. [Corrected by _Antoine Mathys_, May 07 2019]
%C A176499 Farey(n) = A005728(n). [_Franklin T. Adams-Watters_, May 12 2010]
%C A176499 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 (A048211), 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. - _Hugo Pfoertner_, Jan 24 2021
%H A176499 Antoine Mathys, <a href="/A176499/b176499.txt">Table of n, a(n) for n = 1..50</a>
%H A176499 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). Digital Object Identifier (DOI): 10.1119/1.19396.
%H A176499 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 A176499 Sameen Ahmed KHAN, <a href="/A176499/a176499a.nb">Mathematica notebook 1</a>
%H A176499 Sameen Ahmed KHAN, <a href="/A176499/a176499b.nb">Mathematica notebook 2</a>
%H A176499 Hugo Pfoertner, <a href="/plot2a?name1=A048211&amp;name2=A176499&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true&amp;drawlines=true">Ratio for series-parallel networks</a>, Plot2 of A048211(n)/a(n).
%H A176499 Hugo Pfoertner, <a href="/plot2a?name1=A174283&amp;name2=A176499&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true&amp;drawlines=true">Ratio for networks with bridges</a>, Plot2 of A174283(n)/a(n).
%H A176499 Hugo Pfoertner, <a href="/plot2a?name1=A337517&amp;name2=A176499&amp;tform1=untransformed&amp;tform2=untransformed&amp;shift=0&amp;radiop1=ratio&amp;drawpoints=true&amp;drawlines=true">Ratio for arbitrary networks</a>, Plot2 of A337517(n)/a(n).
%F A176499 a(n) = A005728(A000045(n+1)). - _Michel Marcus_, Jul 31 2018
%e A176499 n = 5, m = Fibonacci(5 + 1) = 8, Farey(8) = 23.
%p A176499 with(numtheory): with(combinat,fibonacci): a:=n->1+add(phi(i),i=1..n): seq(a(fibonacci(n+1)),n=1..30); # _Muniru A Asiru_, Jul 31 2018
%t A176499 b[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
%t A176499 a[n_] := b[Fibonacci[n + 1]];
%t A176499 Array[a, 30] (* _Jean-François Alcover_, Sep 20 2018 *)
%o A176499 (PARI) farey(n) = 1+sum(k=1, n, eulerphi(k));
%o A176499 a(n) = farey(fibonacci(n+1)); \\ _Michel Marcus_, Jul 31 2018
%o A176499 (GAP) List([1..30],n->Sum([1..Fibonacci(n+1)],i->Phi(i)))+1; # _Muniru A Asiru_, Jul 31 2018
%o A176499 (Magma) [1+&+[EulerPhi(i):i in [1..Fibonacci(n+1)]]:n in [1..30]]; // _Marius A. Burtea_, Jul 26 2019
%Y A176499 Cf. A000045, A005728.
%Y A176499 Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176500, A176501, A176502, A337517.
%K A176499 nonn
%O A176499 1,1
%A A176499 _Sameen Ahmed Khan_, Apr 21 2010
%E A176499 a(26)-a(29) from _Sameen Ahmed Khan_, May 02 2010
%E A176499 a(30) from _Antoine Mathys_, Aug 06 2018