A137421 Decimal expansion of growth constant in random Fibonacci sequence.

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

Offset: 1

Author

Jonathan Vos Post, Apr 16 2008

Keywords

• easy
• nonn
• cons
• nice

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)

Examples

			1.20556943040059031170202861778382342637710891959769944...
		

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]
• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A078416, A272874. A357468.

Programs

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 *)

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

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)

Extensions

More terms from R. J. Mathar, Apr 23 2008

More terms from Jean-François Alcover, Aug 06 2014