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 A137421 Decimal expansion of growth constant in random Fibonacci sequence. 2
 1, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8, 2, 6, 9, 5, 7, 8, 0, 5, 2, 5 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Real zero of x^3 + x^2 - x - 2. - Charles R Greathouse IV, May 28 2011 This is the infinite nested radical sqrt(1+sqrt(-1+sqrt(1+sqrt(-1+...)))), evaluated as the limit for an increasing (even) number of terms (an odd number of terms gives always 1) and using the main branch of the complex sqrt(z) function. This real-valued constant is in fact the unique attractor of the complex mapping M(z)=sqrt(1+sqrt(-1+z)), with its attraction domain covering the whole complex plane, excluding z = 1, the other invariant point of M(z). Closely related is A272874. - Stanislav Sykora, May 08 2016 LINKS Elise Janvresse, Benoît Rittaud and Thierry De La Rue, Growth rate for the expected value of a generalized random Fibonacci sequence, arXiv:0804.2400 [math.PR], 2008. Benoît Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4. Benoît Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths se font discrètes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). See p. 119. FORMULA In the book by Benoît Rittaud et al. it is stated that this number is cube_root(43/54+sqrt(59/108))+cube_root(43/54-sqrt(59/108))-1/3. - Eric Desbiaux, Sep 13 2008, Oct 17 2008 The largest real solution of x = sqrt(1+sqrt(-1+x)). - Stanislav Sykora, May 08 2016 EXAMPLE alpha - 1 = 1.20556943... where alpha is the only real root of alpha^3 = 2*alpha^2 + 1. This is the growth rate of the expected value of a (1/2,1)-random Fibonacci sequence, defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. The more general case of an unbalanced coin is given by Janvresse, Rittaud and De La Rue. MAPLE Digits := 80 ; fsolve( x^3-2*x^2-1, x, 2.2..2.3)-1.0 ; # R. J. Mathar, Apr 23 2008 MATHEMATICA RealDigits[Root[x^3 + x^2 - x - 2, x, 1], 10, 98] // First (* Jean-François Alcover, Aug 06 2014 *) PROG (PARI) real(polroots(x^3+x^2-x-2)) \\ Charles R Greathouse IV, May 28 2011 (PARI) polrootsreal(x^3+x^2-x-2) \\ Charles R Greathouse IV, May 14 2014 CROSSREFS Cf. A078416, A272874. Sequence in context: A197253 A249693 A251420 * A155524 A051111 A068558 Adjacent sequences:  A137418 A137419 A137420 * A137422 A137423 A137424 KEYWORD easy,nonn,cons AUTHOR Jonathan Vos Post, Apr 16 2008 EXTENSIONS More terms from R. J. Mathar, Apr 23 2008 More terms from Jean-François Alcover, Aug 06 2014 STATUS approved

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Last modified October 20 08:34 EDT 2019. Contains 328253 sequences. (Running on oeis4.)