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A119015 Denominators of "Farey fraction" approximations to e. 5

%I

%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-Fran├žois 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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Last modified May 12 23:35 EDT 2021. Contains 343829 sequences. (Running on oeis4.)