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A003417 Continued fraction for e.
(Formerly M0088)
21
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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

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.

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.

K. Matthews, Finding the continued fraction of e^(l/m)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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

G. Xiao, Contfrac

Index entries for continued fractions for constants

FORMULA

G.f.: (2+x+2x^2-3x^3-x^4+x^6)/(1-2x^3+x^6); a(n)=sum{k=0..n, 2*C(0,k)-C(1,k)-2*sin(2*pi*(k-1)/3)*floor((2k-1)/3)/sqrt(3)} [offset 0]; - Paul Barry, Jun 27 2006

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

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] + 2n/3 - (2n-3)/3*DirichletCharacter[3, 1, n]; Table[a[n], {n, 1, 20}] (* Enrique Pérez Herrero, Feb 23 2013 *)

PROG

(PARI) contfrac(exp(1)-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])); } \\ From Harry J. Smith, Apr 14 2009

(PARI) A003417(n)=if(n%3, 1+(n==1), n\3*2) \\ [M. F. Hasler, May 01 2013]

CROSSREFS

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

Sequence in context: A078997 A024680 A083531 * A158986 A079900 A188317

Adjacent sequences:  A003414 A003415 A003416 * A003418 A003419 A003420

KEYWORD

nonn,cofr,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 22 11:25 EDT 2014. Contains 244815 sequences.