A001204 -id:A001204 - cp's OEIS frontend

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

N. J. A. Sloane

This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey, Aug 07 2004
Sorted with duplicate terms dropped, this is A004277, 1 together with the positive even numbers. - Alonso del Arte, Jan 27 2012
From Peter Bala, Nov 26 2019: (Start)
Related continued fractions expansions:
2*e = [5; 2, 3, 2, 3, 1, 2, 1, 3, 4, 3, 1, 4, 1, 3, 6, 3, 1, 6, ..., 1, 3, 2*n, 3, 1, 2*n, ...].
(1/2)*e = [1; 2, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, 1, 3, 7, 3, 1, 7, ..., 1, 3, 2*n + 1, 3, 1, 2*n + 1, ...].
4*e = [10, 1, 6, 1, 7, 2, 7, 2, 7, 1, 1, 1, 7, 3, 7, 1, 2, 1, 7, 4, 7, 1, 3, 1, 7, 5, 7, 1, 4, ..., 1, 7, n+1, 7, 1, n, ...].
(1/4)*e = [0, 1, 2, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, 7, 1, 3, 2, 1, 1, 1, 4, 7, 1, 4, 2, ..., 1, 1, 1, n, 7, 1, n, 2, ...]. (End)

			2.718281828459... = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...))))

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).

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).

Cf. A001113, A007676, A007677, A001204, A058282, A005131.
Cf. A006083, A006084, A006085, A081750.
Run lengths of A342991.

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

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 *)

contfrac(exp(1)) \\ Alexander R. Povolotsky, Feb 23 2008

{ 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

A003417(n)=if(n%3<>2,1+(n==0),(n+1)/3*2) \\ M. F. Hasler, May 01 2013

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

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

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

Offset changed by Andrew Howroyd, Aug 07 2024

A072334 Decimal expansion of e^2.

Original entry on oeis.org

7, 3, 8, 9, 0, 5, 6, 0, 9, 8, 9, 3, 0, 6, 5, 0, 2, 2, 7, 2, 3, 0, 4, 2, 7, 4, 6, 0, 5, 7, 5, 0, 0, 7, 8, 1, 3, 1, 8, 0, 3, 1, 5, 5, 7, 0, 5, 5, 1, 8, 4, 7, 3, 2, 4, 0, 8, 7, 1, 2, 7, 8, 2, 2, 5, 2, 2, 5, 7, 3, 7, 9, 6, 0, 7, 9, 0, 5, 7, 7, 6, 3, 3, 8, 4, 3, 1, 2, 4, 8, 5, 0, 7, 9, 1, 2, 1, 7, 9

Offset: 1

N. J. A. Sloane, Jul 15 2002

Also where x^(1/sqrt(x)) is a maximum. - Robert G. Wilson v, Oct 22 2014

			7.389056098930650...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 1.4, pages 2 and 28-29.

Harry J. Smith, Table of n, a(n) for n = 1..20000
John Cosgrave, e^2 is irrational.
Index entries for transcendental numbers.

Cf. A001204 (continued fraction).

Cf. A001113, A091933, A092426, A092511, A092512, A092513.

SetDefaultRealField(RealField(100)); Exp(1)^2; // Vincenzo Librandi, Apr 05 2020

RealDigits[E^2, 10, 100][[1]] (* Vincenzo Librandi, Apr 05 2020 *)

default(realprecision, 20080); x=exp(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b072334.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

Equals Sum_{n>=0} Sum_{k>=0} 1/(n!*k!). - Fredrik Johansson, Apr 21 2006
Equals Sum_{n>=0} 2^n/n!. - Daniel Hoyt Nov 20 2020
From Peter Bala, Jan 13 2022: (Start)
e^2 = Sum_{n >= 0} 2^n/n!. Faster converging series include
e^2 = 8*Sum_{n >= 0} 2^n/(p(n-1)*p(n)*n!), where p(n) = n^2 - n + 2 and
e^2 = -48*Sum_{n >= 0} 2^n/(q(n-1)*q(n)*n!), where q(n) = n^3 + 5*n - 2.
e^2 = 7 + Sum_{n >= 0} 2^(n+3)/((n+2)^2*(n+3)^2*n!) and
7/e^2 = 1 - Sum_{n >= 0} (-2)^(n+1)*n^2/(n+2)!.
e^2 = 7 + 2/(5  + 1/(7 + 1/(9  + 1/(11 + ...)))) (follows from the fact that A004273 is the continued fraction expansion of tanh(1) = (e^2 - 1)/ (e^2 + 1)). Cf. A001204. (End)
Equals lim_{n->oo} (Sum_{k=1..n} 1/binomial(n,k)^x)^(n^x), for all real x > 1/2 (Furdui, 2013). - Amiram Eldar, Mar 26 2022

A058282 Continued fraction for e^3.

Original entry on oeis.org

20, 11, 1, 2, 4, 3, 1, 5, 1, 2, 16, 1, 1, 16, 2, 13, 14, 4, 6, 2, 1, 1, 2, 2, 2, 3, 5, 1, 3, 1, 1, 68, 7, 5, 1, 4, 2, 1, 1, 1, 1, 1, 1, 7, 3, 1, 6, 1, 2, 5, 4, 7, 2, 1, 3, 2, 2, 1, 2, 1, 4, 1, 1, 13, 1, 1, 2, 1, 1, 1, 1, 3, 7, 11, 18, 54, 1, 2, 2, 2, 1, 1, 6, 2, 2, 46, 2, 189, 1, 24, 1, 8, 13, 4, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

			20.085536923187667740928529... = 20 + 1/(11 + 1/(1 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, Apr 30 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
K. Matthews, Finding the continued fraction of e^(l/m) ["... there is no known formula for the partial quotients of the continued fraction expansion of e^3, or more generally e^(l/m) with l distinct from 1,2 and gcd(l,m)=1..."]
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A001204, A003417, A005131.

with(numtheory); Digits:=200: cf:=convert(evalf( exp(3)), confrac); # N. J. A. Sloane, Sep 05 2012

Mathematica
```
ContinuedFraction[ E^3, 100]
```
PARI
```
contfrac(exp(1)^3)
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(3)); for (n=1, 20001, write("b058282.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 30 2009

More terms from Jason Earls, Jul 10 2001

A159824 Continued fraction for Pi^Pi (cf. A073233).

Original entry on oeis.org

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 0, 0, 3, 1, 1, 3, 0, 6, 4, 7, 5, 2, 9, 9, 8, 10, 8, 9, 1, 13, 18, 1, 2, 8, 15, 26, 10, 22, 1, 18, 9, 20, 10, 2, 6, 13, 19, 16, 38, 38, 3, 32, 5, 39, 24, 7, 27, 14, 41, 20, 39, 32, 7, 20, 35, 44, 50, 24, 34, 51, 14, 39, 47, 49, 15, 61, 54, 60, 52, 34, 60, 32, 72, 48, 12, 67, 52, 22, 48

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/exp(2) = 0 + 0/2! + 0/3! + 3/4! + 1/5! + 1/6! + 3/7! + 0/8! + 6/9! +...

Index entries for factorial base representation

Cf. A092553 (decimal expansion), 0 U A001204 (continued fraction).

Cf. A054977 (e), A067840 (e^2), A068453 (sqrt(e)), A237420 (1/e).

SetDefaultRealField(RealField(250));  [Floor(Exp(-2))] cat [Floor(Factorial(n)*Exp(-2)) - n*Floor(Factorial((n-1))*Exp(-2)) : n in [2..80]];

With[{b = 1/E^2}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(-2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=exp(-2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

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A003417 Continued fraction for e.

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A072334 Decimal expansion of e^2.

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A058282 Continued fraction for e^3.

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A159824 Continued fraction for Pi^Pi (cf. A073233).

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A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

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