cp's OEIS Frontend

A007676 Numerators of convergents to e.

%I A007676 M0869 #71 Jul 26 2025 03:32:55
%S A007676 2,3,8,11,19,87,106,193,1264,1457,2721,23225,25946,49171,517656,
%T A007676 566827,1084483,13580623,14665106,28245729,410105312,438351041,
%U A007676 848456353,14013652689,14862109042,28875761731,534625820200,563501581931,1098127402131,22526049624551
%N A007676 Numerators of convergents to e.
%C A007676 Same as A113873 without its first two terms. - _Jonathan Sondow_, Aug 16 2006
%D A007676 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
%D A007676 W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
%D A007676 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007676 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 A007676 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 A007676 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 A007676 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 A007676 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 A007676 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 A007676 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 A007676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/eContinuedFraction.html">e Continued Fraction</a>
%H A007676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SultansDowryProblem.html">Sultan's Dowry Problem</a>
%e A007676 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 A007676 Digits := 60: convert(evalf(E),confrac,50,'cvgts'): cvgts;
%t A007676 Numerator[Convergents[E, 30]] (* _T. D. Noe_, Oct 12 2011 *)
%t A007676 Numerator[Table[Piecewise[{
%t A007676    {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0},
%t A007676    {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 A007676    {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 A007676 }], {n, 0, 30}]] (* _Eric W. Weisstein_, Sep 09 2013 *)
%t A007676 Table[Piecewise[{
%t A007676    {(-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 A007676    {((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1], Mod[n, 3] == 1},
%t A007676    {(5/3 + (2 n)/3)!/((1 + n)/3)! Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1], Mod[n, 3] == 2}
%t A007676 }], {n, 0, 30}] (* _Eric W. Weisstein_, Sep 10 2013 *)
%Y A007676 Cf. A007677 (denominators of convergents to e).
%Y A007676 Cf. A003417 (continued fraction of e).
%K A007676 nonn,easy,nice,frac
%O A007676 0,1
%A A007676 _N. J. A. Sloane_, _Robert G. Wilson v_