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A007676 Numerators of convergents to e.
(Formerly M0869)
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Same as A113873 without its first two terms. - Jonathan Sondow, Aug 16 2006

REFERENCES

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.

LINKS

Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)

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

EXAMPLE

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, ...

MAPLE

Digits := 60: convert(evalf(E), confrac, 50, 'cvgts'): cvgts;

MATHEMATICA

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 *)

CROSSREFS

Cf. A007677 (denominators of convergents to e).

Cf. A003417 (continued fraction of e).

Sequence in context: A041893 A206241 A113873 * A042443 A042263 A153439

Adjacent sequences:  A007673 A007674 A007675 * A007677 A007678 A007679

KEYWORD

nonn,easy,nice,frac

AUTHOR

N. J. A. Sloane, Robert G. Wilson v

STATUS

approved

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Last modified November 20 17:58 EST 2014. Contains 249754 sequences.