

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
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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 strict upper bound provided by this difficult to compute (due to computer memory) sequence is better than the easier to compute Sequence A176499.


LINKS

Table of n, a(n) for n=1..27.
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(v=List(0), m=fibonacci(n + 1)); for(b=n+1, m, for(a=1, (b1)/n, listput(v, a/b))); my(l=length(vecsort(v, , 8))); farey(m)  l; \\ Antoine Mathys, Aug 08 2018


CROSSREFS

Cf. A048211, A153588, A174283, A174284, A174285 and A174286, A176499, A176500, A176502.
Sequence in context: A000080 A153447 A076893 * A096354 A076838 A034749
Adjacent sequences: A176498 A176499 A176500 * A176502 A176503 A176504


KEYWORD

more,nonn


AUTHOR

Sameen Ahmed Khan, Apr 21 2010


EXTENSIONS

a(19)a(27) from Antoine Mathys, Aug 10 2018


STATUS

approved



