A003417 Continued fraction for e.
2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 1, 1, 24, 1, 1, 26, 1, 1, 28, 1, 1, 30, 1, 1, 32, 1, 1, 34, 1, 1, 36, 1, 1, 38, 1, 1, 40, 1, 1, 42, 1, 1, 44, 1, 1, 46, 1, 1, 48, 1, 1, 50, 1, 1, 52, 1, 1, 54, 1, 1, 56, 1, 1, 58, 1, 1, 60, 1, 1, 62, 1, 1, 64, 1, 1, 66
Offset: 0
Examples
2.718281828459... = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...))))
References
- John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
- CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.2.
- Jay R. Goldman, The Queen of Mathematics, 1998, p. 70.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..9999
- Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, Continued Fractions with Predictable Patterns and Transcendental Numbers, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.4.
- Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
- Henry Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. [JSTOR] and arXiv:math/0601660 [math.NT], 2006.
- Stephen Crowley, Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, and Fractal Strings, vixra:1202.0079 v2, 2012. [Wayback Machine link]
- Francesco Dolce and Pierre-Adrien Tahay, Column representation of Sturmian words in cellular automata, Czech Technical University (Prague, Czechia, 2022).
- W. R. Harmon, Letter to N. J. A. Sloane, Sep 1974
- MathOverflow, What is the effect of adding 1/2 to a continued fraction?, 2013.
- Keith Matthews, Finding the continued fraction of e^(l/m)
- Sophie Morier-Genoud and Valentin Ovsienko, On q-deformed real numbers, arXiv:1908.04365 [math.QA], 2019.
- C. D. Olds, The simple continued fraction expansion of e, Am. Math. Monthly 77 (9) (1970) 968-974.
- Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
- Oskar Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 134.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Ward Whitt, Weirdness in CTMC's, Notes for Course IEOR 6711: Stochastic Models I, [PDF], 2012. - From _N. J. A. Sloane_, Jan 03 2013
- Eric Weisstein's World of Mathematics, e Continued Fraction.
- Gang Xiao, Contfrac.
- Yaohui Zhu, Kaiming Sun, Zhengdong Luo, and Lingfeng Wang, Progressive Self-Learning for Domain Adaptation on Symbolic Regression of Integer Sequences, Proc. 39th AAAI Conf. Artif. Intel. (2025) Vol. 39, No. 1, 1692-1699. See p. 1698.
- Index entries for continued fractions for constants.
- Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Crossrefs
Programs
-
Maple
numtheory[cfrac](exp(1),100,'quotients'); # Jani Melik, May 25 2006 A003417:=(2+z+2*z**2-3*z**3-z**4+z**6)/(z-1)**2/(z**2+z+1)**2; # Simon Plouffe in his 1992 dissertation
-
Mathematica
ContinuedFraction[E, 100] (* Stefan Steinerberger, Apr 07 2006 *) a[n_] := KroneckerDelta[1, n] + 2 n/3 - (2 n - 3)/3 DirichletCharacter[3, 1, n]; Table[a[n], {n, 1, 20}] (* Enrique Pérez Herrero, Feb 23 2013 *) Table[Piecewise[{{2, n == 0}, {2 (n + 1)/3, Mod[n, 3] == 2}}, 1], {n, 0, 120}] (* Eric W. Weisstein, Jan 05 2019 *) Join[{2}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 2, 1, 1, 4, 1}, 120]] (* Eric W. Weisstein, Jan 05 2019 *) Join[{2}, Table[(2 (n + 4) + (1 - 2 n) Cos[2 n Pi/3] + Sqrt[3] (1 - 2 n) Sin[2 n Pi/3])/9, {n, 120}]] (* Eric W. Weisstein, Jan 05 2019 *) Join[{2}, Flatten[Table[{1, 2n, 1}, {n, 40}]]] (* Harvey P. Dale, Jan 21 2020 *)
-
PARI
contfrac(exp(1)) \\ Alexander R. Povolotsky, Feb 23 2008
-
PARI
{ allocatemem(932245000); default(realprecision, 25000); x=contfrac(exp(1)); for (n=1, 10000, write("b003417.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 14 2009 -
PARI
A003417(n)=if(n%3<>2,1+(n==0),(n+1)/3*2) \\ M. F. Hasler, May 01 2013
-
Python
def A003417(n): return 2 if n == 0 else 1 if n % 3 != 2 else (n+1)//3<<1 # Chai Wah Wu, Jul 27 2022
-
Scala
def eContFracTrio(n: Int): List[Int] = List(1, 2 * n, 1) 2 +: ((1 to 40).map(eContFracTrio).flatten) // Alonso del Arte, Nov 22 2020, with thanks to Harvey P. Dale
Formula
From Paul Barry, Jun 27 2006: (Start)
G.f.: (2 + x + 2*x^2 - 3*x^3 - x^4 + x^6)/(1 - 2*x^3 + x^6).
a(n) = 0^n + Sum{k = 0..n} 2*sin(2*Pi*(k - 1)/3)*floor((2*k - 1)/3)/sqrt(3). [Corrected and simplified by Jianing Song, Jan 05 2019] (End)
a(n) = 2*a(n-3) - a(n-6), n >= 7. - Philippe Deléham, Feb 10 2009
G.f.: 1 + U(0) where U(k)= 1 + x/(1 - x*(2*k + 1)/(1 + x*(2*k + 1) - 1/((2*k + 1) + 1 - (2*k + 1)*x/(x + 1/U(k+1))))); (continued fraction, 5-step). - Sergei N. Gladkovskii, Oct 07 2012
a(3*n-1) = 2*n, a(0) = 2, a(n) = 1 otherwise (i.e., for n+1 > 1, not a multiple of 3). - M. F. Hasler, May 01 2013
E.g.f.: First derivative of (2/9)*exp(x)*(x + 3) + (2/9)*exp(-x/2)*(2*x*cos((sqrt(3)/2)*x+2*Pi/3) - 3*cos((sqrt(3)/2)*x)) + x. - Jianing Song, Jan 05 2019
a(n) = floor(1/(n+1))-(floor(n/3)-floor((n+1)/3))*(2*n-1)/3+1. - Aaron J Grech, Sep 06 2024
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - log(2)/2. - Amiram Eldar, May 03 2025
Extensions
Offset changed by Andrew Howroyd, Aug 07 2024
Comments