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 A003417 Continued fraction for e. (Formerly M0088) 26
 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. 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, C Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445, 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. 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. W. R. Harmon, Letter to N. J. A. Sloane, Sep 1974 K. Matthews, Finding the continued fraction of e^(l/m) 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. 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 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. [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])); } \\ 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 STATUS approved

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Last modified December 18 18:34 EST 2018. Contains 318243 sequences. (Running on oeis4.)