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 A003417 Continued fraction for e. (Formerly M0088) 37
 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 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. 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. 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. 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. Run lengths of A342991. Sequence in context: A078997 A024680 A083531 * A358549 A158986 A079900 Adjacent sequences: A003414 A003415 A003416 * A003418 A003419 A003420 KEYWORD nonn,cofr,nice,easy,changed AUTHOR N. J. A. Sloane STATUS approved

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Last modified July 13 09:39 EDT 2024. Contains 374274 sequences. (Running on oeis4.)