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A176501 a(n) = Farey(m; I) where m = Fibonacci(n + 1) and I = [1/n, 1].

%I A176501 #68 Sep 01 2025 23:43:48
%S A176501 1,2,4,9,19,50,122,317,837,2213,5758,15236,40028,105079,276627,727409,
%T A176501 1910685,5020094,13180380,34600740,90814431,238288480,625111687,
%U A176501 1639676484,4300183922,11275936787,29564497466,77507123132,203175049457,532552499826,1395790412496
%N A176501 a(n) = Farey(m; I) where m = Fibonacci(n + 1) and I = [1/n, 1].
%C A176501 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]
%C A176501 From _Hugo Pfoertner_, Jan 24 2021: (Start)
%C A176501 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, 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.
%C A176501 But in contrast to A176499, which at least correctly bounds A048211, the terms a(5), ..., a(9) in this sequence are smaller than the corresponding terms from A048211 (a(n) vs. A048211(n): 19/22, 50/53, 122/131, 317/337, 837/869). (End)
%H A176501 Antoine Mathys, <a href="/A176501/b176501.txt">Table of n, a(n) for n = 1..40</a>
%H A176501 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 A176501 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 A176501 Sameen Ahmed Khan, <a href="/A176501/a176501.nb">Mathematica notebook</a>
%H A176501 Hugo Pfoertner, <a href="/plot2a?name1=A048211&amp;name2=A176501&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 A176501 Hugo Pfoertner, <a href="/plot2a?name1=A174283&amp;name2=A176501&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 A176501 Hugo Pfoertner, <a href="/plot2a?name1=A337517&amp;name2=A176501&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).
%e A176501 n = 5, I = [1/5, 1], m = Fibonacci(5 + 1) = 8, Farey(8) = 23, Farey(8; I) = 19
%t A176501 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]];
%t A176501 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* _Jean-François Alcover_, Aug 30 2018, after _Antoine Mathys_ *)
%o A176501 (PARI) farey(n) = sum(i=1, n, eulerphi(i)) + 1;
%o A176501 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
%Y A176501 Cf. A048211, A153588, A174283, A174284, A174285, A174286, A176499, A176500, A176502, A337517.
%K A176501 nonn,changed
%O A176501 1,2
%A A176501 _Sameen Ahmed Khan_, Apr 21 2010
%E A176501 a(19)-a(27) from _Antoine Mathys_, Aug 10 2018
%E A176501 a(28)-a(31) from _Antoine Mathys_, May 07 2019