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A007676
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Numerators of convergents to e.
(Formerly M0869)
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16
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2, 3, 8, 11, 19, 87, 106, 193, 1264, 1457, 2721, 23225, 25946, 49171, 517656, 566827, 1084483, 13580623, 14665106, 28245729, 410105312, 438351041, 848456353, 14013652689, 14862109042, 28875761731, 534625820200, 563501581931, 1098127402131, 22526049624551
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OFFSET
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0,1
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COMMENTS
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Same as A113873 without its first two terms. - Jonathan Sondow, Aug 16 2006
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REFERENCES
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CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
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LINKS
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Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers , J. Int. Seq. 14 (2011) # 11.1.4
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Eric Weisstein's World of Mathematics, e Continued Fraction
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem
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EXAMPLE
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2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, 517656/190435, 566827/208524, 1084483/398959, 13580623/4996032, 14665106/5394991, 28245729/10391023, 410105312/150869313, 438351041/161260336, 848456353/312129649, ...
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MAPLE
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Digits := 60: convert(evalf(E), confrac, 50, 'cvgts'): cvgts;
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MATHEMATICA
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Numerator[Convergents[E, 30]] (* T. D. Noe, Oct 12 2011 *)
Numerator[Table[Piecewise[{
{Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
{Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},
{Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1]/Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}
}], {n, 0, 30}]] (* Eric W. Weisstein, Sep 09 2013 *)
Table[Piecewise[{
{(-1 + (2 (3 + n))/3)!/(-1 + (3 + n)/3)! Hypergeometric1F1[1/3 (-3 - n), 1 - (2 (3 + n))/3, 1], Mod[n, 3] == 0},
{((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1], Mod[n, 3] == 1},
{(5/3 + (2 n)/3)!/((1 + n)/3)! Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1], Mod[n, 3] == 2}
}], {n, 0, 30}] (* Eric W. Weisstein, Sep 10 2013 *)
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CROSSREFS
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Cf. A007677 (denominators of convergents to e).
Cf. A003417 (continued fraction of e).
Sequence in context: A206241 A295333 A113873 * A042443 A042263 A280323
Adjacent sequences: A007673 A007674 A007675 * A007677 A007678 A007679
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KEYWORD
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nonn,easy,nice,frac
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AUTHOR
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N. J. A. Sloane, Robert G. Wilson v
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STATUS
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approved
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