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

Author

Joshua Zucker, May 08 2006

Keywords

• easy
• frac
• nonn

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, ...

Examples

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

Links

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

Crossrefs

For another version see A006258.

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

Programs

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 *)