login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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 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)[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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 30 18:30 EDT 2017. Contains 284302 sequences.