A119014 Numerators of "Farey fraction" approximations to e.

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

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-François 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