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%I #8 May 07 2016 12:30:32
%S 0,1,1,1,1,2,3,4,7,11,18,25,32,39,71,110,181,252,323,394,465,536,1001,
%T 1537,2538,3539,4540,5541,6542,7543,8544,9545,18089,27634,45723,63812,
%U 81901,99990,118079,136168,154257,172346,190435,208524,398959,607483
%N Denominators of "Farey fraction" approximations to e.
%C "Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1, 3/1. Now 3/1 is too big, so add 2/1 to make the fraction smaller: 5/2, 8/3, 11/4. Now 11/4 is too small, so add 8/3 to make the fraction bigger: 19/7, ...
%H Dave Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/99/farey">Farey fractions on sci.math</a> [Broken link]
%H Dave Rusin, <a href="/A002965/a002965.txt">Farey fractions on sci.math</a> [Cached copy]
%e The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 5/2, 8/3, 11/4, 19/7, ...
%t f[x_, n_] := (m = Floor[x]; f0 = {m, m+1/2, m+1}; r = ({a___, b_, c_, d___} /; b < x < c) :> {b, (Numerator[b] + Numerator[c]) / (Denominator[b] + Denominator[c]), c};
%t Join[{m, m+1}, NestList[# /. r &, f0, n-3][[All, 2]]]);
%t Join[{0, 1, 1}, f[E, 43] // Denominator]
%t (* _Jean-François Alcover_, May 18 2011 *)
%Y For another version see A006259.
%Y Cf. A097545, A097546 gives the similar sequence for pi. A119014 gives the numerators for this sequence.
%K easy,frac,nonn
%O 0,6
%A _Joshua Zucker_, May 08 2006
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