A119014
Numerators of "Farey fraction" approximations to e.
Original entry on oeis.org
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
The fractions are 1/0, 0/1, 1/1, 2/1, 3/1, 5/2, 8/3, 11/4, 19/7, ...
Cf.
A097545,
A097546 gives the similar sequence for pi.
A119015 gives the denominators for this sequence.
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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 *)
A006259
Denominators of approximations to e.
Original entry on oeis.org
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: 1
A006258/A006259 = 1, 2, 3, 5/2, 8/3, 11/4, 19/7, 30/11, 49/18, 68/25, 87/32, 106/39, 193/71, 299/110, 492/181, ... .
- R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 122.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A227777
Least splitter of n-th and (n+1)st partial sums of 1/0! + 1/1! + ... + 1/n! + ... = e.
Original entry on oeis.org
1, 2, 3, 7, 39, 110, 252, 465, 1001, 9545, 27634, 136168, 589394, 398959, 5394991, 36568060, 130087267, 312129649, 5779594018, 5467464369, 69204258903, 186055048882, 403978495031, 8690849042711, 25668568633102, 246378923308185, 1163579759684330
Offset: 1
The first 19 splitting rationals are 2, 5/2, 8/3, 19/7, 106/39, 299/110, 685/252, 1264/465, 2721/1001, 25946/9545, 75117/27634, 370143/136168, 1602139/589394, 1084483/398959, 14665106/5394991, 99402293/36568060, 353613854/130087267, 848456353/312129649 & 15710565395/5779594018. Regarding the last one, |15710565395/5779594018 - e| < 10^(-19).
The numerators of these rationals are a proper subsequence of A006258 & A119014 and the denominators are a proper subsequence of A006259 & A119015. - _Robert G. Wilson v_, Jun 27 2015
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z = 16; r[x_, y_] := Module[{a, b, x1 = Min[{x, y}], y1 = Max[{x, y}]}, If[x == y, x, b = NestWhile[#1 + 1 &, 1, ! (a = Ceiling[#1 x1 - 1]) < Ceiling[#1 y1] - 1 &]; (a + 1)/b]]; s[n_] := s[n] = Sum[1/(k - 1)!, {k, 1, n}]; N[Table[s[k], {k, 1, z}]]; t = Table[r[s[n], s[n + 1]], {n, 2, z}]; fd = Denominator[t] (* Peter J. C. Moses, Jul 20 2013 *)
Showing 1-3 of 3 results.
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