A119015 Denominators of "Farey fraction" approximations to e.

0, 1, 1, 1, 1, 2, 3, 4, 7, 11, 18, 25, 32, 39, 71, 110, 181, 252, 323, 394, 465, 536, 1001, 1537, 2538, 3539, 4540, 5541, 6542, 7543, 8544, 9545, 18089, 27634, 45723, 63812, 81901, 99990, 118079, 136168, 154257, 172346, 190435, 208524, 398959, 607483

Offset

0,6

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..45.

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[{0, 1, 1}, f[E, 43] // Denominator]
(* Jean-François Alcover, May 18 2011 *)

Crossrefs

For another version see A006259.

Cf. A097545, A097546 gives the similar sequence for pi. A119014 gives the numerators for this sequence.

Sequence in context: A060987 A359743 A006259 * A222329 A018145 A050195

Adjacent sequences: A119012 A119013 A119014 * A119016 A119017 A119018

Keyword

easy,frac,nonn

Author

Joshua Zucker, May 08 2006

Status

approved