

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
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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 realvalued 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

Table of n, a(n) for n=1..98.
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 JeanChristophe Novelli, Quand les maths se font discrètes, Le Pommier, 2008 (ISBN 9782746503700). 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(43/54sqrt(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_{n1} +/ g_{n2} , 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^32*x^21, 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 (* JeanFrançois Alcover, Aug 06 2014 *)


PROG

(PARI) real(polroots(x^3+x^2x2)[1]) \\ Charles R Greathouse IV, May 28 2011
(PARI) polrootsreal(x^3+x^2x2)[1] \\ 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 JeanFrançois Alcover, Aug 06 2014


STATUS

approved



