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A007677 Denominators of convergents to e.
(Formerly M2343)
21
1, 1, 3, 4, 7, 32, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959, 4996032, 5394991, 10391023, 150869313, 161260336, 312129649, 5155334720, 5467464369, 10622799089, 196677847971, 207300647060, 403978495031, 8286870547680, 8690849042711 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Same as A113874 without its first two terms. - Jonathan Sondow, Aug 16 2006

REFERENCES

E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

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: E := exp(1); convert(evalf(E), confrac, 50, 'cvgts'): cvgts;

MATHEMATICA

Denominator[Convergents[E, 40]] (* T. D. Noe, Oct 12 2011 *)

Denominator[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 n)/3)!/(n/3)! Hypergeometric1F1[-(n/3), -1 - (2 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},

    {((4 + n) (5/3 + (2 n)/3)! )/(3 ((4 + n)/3)!) Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2}

}], {n, 0, 30}]  (* Eric W. Weisstein, Sep 10 2013 *)

CROSSREFS

Cf. A007676 (numerators of convergents to e).

Cf. A003417 (continued fraction of e).

Sequence in context: A270373 A117764 A113874 * A042773 A042173 A046461

Adjacent sequences:  A007674 A007675 A007676 * A007678 A007679 A007680

KEYWORD

nonn,easy,nice,frac

AUTHOR

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

STATUS

approved

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Last modified December 7 11:34 EST 2016. Contains 278874 sequences.