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A007676 Numerators of convergents to e.
(Formerly M0869)
16

%I M0869

%S 2,3,8,11,19,87,106,193,1264,1457,2721,23225,25946,49171,517656,

%T 566827,1084483,13580623,14665106,28245729,410105312,438351041,

%U 848456353,14013652689,14862109042,28875761731,534625820200,563501581931,1098127402131,22526049624551

%N Numerators of convergents to e.

%C Same as A113873 without its first two terms. - _Jonathan Sondow_, Aug 16 2006

%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.

%D W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Eric W. Weisstein, <a href="/A007676/b007676.txt">Table of n, a(n) for n = 0..1000</a> (first 200 terms from T. D. Noe)

%H L. Bayon, P. Fortuny, J. M. Grau, M. M. Ruiz, M. A. Oller-Marcen, <a href="https://arxiv.org/abs/1809.06390">The Best-or-Worst and the Postdoc problems with random number of candidates</a>, arXiv:1809.06390 [math.PR], 2018.

%H C. Elsner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Elsner/elsner9.html">Series of Error Terms for Rational Approximations of Irrational Numbers </a>, J. Int. Seq. 14 (2011) # 11.1.4.

%H C. Elsner, M. Stein, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Elsner2/elsner10.html">On Error Sum Functions Formed by Convergents of Real Numbers</a>, J. Int. Seq. 14 (2011) # 11.8.6.

%H J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.

%H J. Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.

%H J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/eContinuedFraction.html">e Continued Fraction</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>

%e 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, ...

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

%t Numerator[Convergents[E, 30]] (* _T. D. Noe_, Oct 12 2011 *)

%t Numerator[Table[Piecewise[{

%t {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},

%t {Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1},

%t {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}

%t }], {n, 0, 30}]] (* _Eric W. Weisstein_, Sep 09 2013 *)

%t Table[Piecewise[{

%t {(-1 + (2 (3 + n))/3)!/(-1 + (3 + n)/3)! Hypergeometric1F1[1/3 (-3 - n), 1 - (2 (3 + n))/3, 1], Mod[n, 3] == 0},

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

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

%t }], {n, 0, 30}] (* _Eric W. Weisstein_, Sep 10 2013 *)

%Y Cf. A007677 (denominators of convergents to e).

%Y Cf. A003417 (continued fraction of e).

%K nonn,easy,nice,frac

%O 0,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_

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Last modified February 23 07:11 EST 2020. Contains 332159 sequences. (Running on oeis4.)