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A137421 Decimal expansion of growth constant in random Fibonacci sequence. 5
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
From Wolfdieter Lang, Oct 17 2022: (Start)
This equals r0 - 1/3 where r0 is the real root of y^3 - (4/3)*y - 43/27.
The other roots of x^3 + x^2 - x - 2 are (w1*(4*(43 + 3*sqrt(177)))^(1/3) + w2*(4*(43 - 3*sqrt(177)))^(1/3) - 2)/6 = -1.1027847152... + 0.6654569511...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1 + 2*cosh((1/3)*arccosh(43/16)) - 2*sqrt(3)*sinh((1/3)*arccosh(43/16))*i)/3, and its complex conjugate.
(End)
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. [Broken link]
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
From Wolfdieter Lang, Oct 17 2022: (Start)
Equals ((4*(43 + 3*sqrt(177)))^(1/3) + 16*(4*(43 + 3*sqrt(177)))^(-1/3) - 2)/6.
Equals ((4*(43 + 3*sqrt(177)))^(1/3) + (4*(43 - 3*sqrt(177)))^(1/3) - 2)/6.
Equals (4*cosh((1/3)*arccosh(43/16)) - 1)/3. (End)
EXAMPLE
1.20556943040059031170202861778382342637710891959769944...
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)[1]) \\ Charles R Greathouse IV, May 28 2011
(PARI) polrootsreal(x^3+x^2-x-2)[1] \\ Charles R Greathouse IV, May 14 2014
CROSSREFS
Sequence in context: A249693 A251420 A361582 * A155524 A051111 A068558
KEYWORD
easy,nonn,cons,nice
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 March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)