A003417 Continued fraction for e. (Formerly M0088)

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

1,1

Comments

This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey, Aug 07 2004

First differences are A120691. - Paul Barry, Jun 27 2006

Sorted with duplicate terms dropped, this is A004277, 1 together with the positive even numbers. - Alonso del Arte, Jan 27 2012

Run lengths of A342991. - Paolo Xausa, Aug 26 2021

References

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

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.2.

J. R. Goldman, The Queen of Mathematics, 1998, p. 70.

O. Perron, Die Lehre von den Kettenbrüchen, 2nd ed., Teubner, Leipzig, 1929, p. 134.

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 = 1..10000

Thomas Baruchel and C. Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.

H. 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.

S. Crowley, Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings, vixra:1202.0079 v2, 2012.

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?

K. 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.

T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.

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. O. 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

Index entries for continued fractions for constants

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

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) [with offset 0]. [Corrected and simplified by Jianing Song, Jan 05 2019] (End)

a(n) = 2*a(n-3) - a(n-6), n >= 8. - 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) = 2*n, a(1) = 2, a(n) = 1 otherwise (i.e., for n > 1, not a multiple of 3). - M. F. Hasler, May 01 2013

E.g.f.: (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

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)

Example

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

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 *)

Prog

(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, " ", x[n])); } \\ Harry J. Smith, Apr 14 2009
(PARI) A003417(n)=if(n%3, 1+(n==1), n\3*2) \\ M. F. Hasler, May 01 2013
(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
(Python)
def A003417(n): return 2 if n == 1 else 1 if n % 3 else n//3<<1 # Chai Wah Wu, Jul 27 2022

Crossrefs

Cf. A001113, A007676, A007677, A001204, A058282, A005131.

Cf. A006083, A006084, A006085, A081750.

Sequence in context: A078997 A024680 A083531 * A358549 A158986 A079900

Adjacent sequences: A003414 A003415 A003416 * A003418 A003419 A003420

Keyword

nonn,cofr,nice,easy

Author

N. J. A. Sloane

Status

approved