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A119014 Numerators of "Farey fraction" approximations to e. 5
1, 0, 1, 2, 3, 5, 8, 11, 19, 30, 49, 68, 87, 106, 193, 299, 492, 685, 878, 1071, 1264, 1457, 2721, 4178, 6899, 9620, 12341, 15062, 17783, 20504, 23225, 25946, 49171, 75117, 124288, 173459, 222630, 271801, 320972, 370143, 419314, 468485, 517656, 566827 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

"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, ...

LINKS

Table of n, a(n) for n=0..43.

Dave Rusin, Farey fractions on sci.math [Broken link]

Dave Rusin, Farey fractions on sci.math [Cached copy]

EXAMPLE

The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 5/2, 8/3, 11/4, 19/7, ...

MATHEMATICA

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};

Join[{m, m + 1}, NestList[# /. r &, f0, n - 3][[All, 2]]]); Join[{1, 0, 1 }, f[E, 41]] // Numerator

(* Jean-Fran├žois Alcover, May 18 2011 *)

CROSSREFS

For another version see A006258.

Cf. A097545, A097546 gives the similar sequence for pi. A119015 gives the denominators for this sequence.

Sequence in context: A004693 A272136 A254308 * A006258 A177967 A265741

Adjacent sequences:  A119011 A119012 A119013 * A119015 A119016 A119017

KEYWORD

easy,frac,nonn

AUTHOR

Joshua Zucker, May 08 2006

STATUS

approved

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Last modified April 18 05:44 EDT 2021. Contains 343072 sequences. (Running on oeis4.)