

A176501


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


12



1, 2, 4, 9, 19, 50, 122, 317, 837, 2213, 5758, 15236, 40028, 105079, 276627, 727409, 1910685, 5020094, 13180380, 34600740, 90814431, 238288480, 625111687, 1639676484, 4300183922, 11275936787, 29564497466, 77507123132, 203175049457, 532552499826, 1395790412496
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OFFSET

1,2


COMMENTS

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 nonplanar) 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]


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), 175179 (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.genph], (20 April 2010).
Sameen Ahmed Khan, Mathematica notebook


EXAMPLE

n = 5, I = [1/5, 1], m = Fibonacci(5 + 1) = 8, Farey(8) = 23, Farey(8; I) = 19


MATHEMATICA

a[n_ /; n<4] := 2^(n1); a[n_] := Module[{m = Fibonacci[n+1], v}, v = Reap[ Do[Sow[j/i], {i, n+1, m}, {j, 1, (i1)/n}]][[2, 1]]; Total[ EulerPhi[ Range[m]]]  Length[v // Union]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 23}] (* JeanFrançois Alcover, Aug 30 2018, after Antoine Mathys *)


PROG

(PARI) farey(n) = sum(i=1, n, eulerphi(i)) + 1;
a(n) = my(m=fibonacci(n + 1), count=0); for(b=n+1, m, for(a=1, (b1)/n, if(gcd(a, b)==1, count++))); farey(m)  1  count; \\ Antoine Mathys, May 07 2019


CROSSREFS

Cf. A048211, A153588, A174283, A174284, A174285 and A174286, A176499, A176500, A176502.
Sequence in context: A327017 A153447 A076893 * A324317 A096354 A076838
Adjacent sequences: A176498 A176499 A176500 * A176502 A176503 A176504


KEYWORD

nonn


AUTHOR

Sameen Ahmed Khan, Apr 21 2010


EXTENSIONS

a(19)a(27) from Antoine Mathys, Aug 10 2018
a(28)a(31) from Antoine Mathys, May 07 2019


STATUS

approved



