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A119014
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Numerators of "Farey fraction" approximations to e.
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4
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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; internal format)
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OFFSET
| 0,4
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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, ...
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LINKS
| Dave Rusin, Farey fractions on sci.math
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EXAMPLE
| The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 5/2, 8/3, 11/4, 19/7, ...
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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
(* From Jean-François Alcover, May 18 2011 *)
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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: A065462 A062762 A004693 * A006258 A177967 A114831
Adjacent sequences: A119011 A119012 A119013 * A119015 A119016 A119017
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 08 2006
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