A137421 - OEIS

%I #62 Jan 05 2026 16:51:35

%S 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,

%T 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,

%U 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

%N Decimal expansion of growth constant in random Fibonacci sequence.

%C Real zero of x^3 + x^2 - x - 2. - _Charles R Greathouse IV_, May 28 2011

%C 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

%C From _Wolfdieter Lang_, Oct 17 2022: (Start)

%C This equals r0 - 1/3 where r0 is the real root of y^3 - (4/3)*y - 43/27.

%C 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.

%C 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.

%C (End)

%H Elise Janvresse, Benoît Rittaud and Thierry De La Rue, <a href="http://arXiv.org/abs/0804.2400">Growth rate for the expected value of a generalized random Fibonacci sequence</a>, arXiv:0804.2400 [math.PR], 2008.

%H Benoît Rittaud, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rittaud2/rittaud11.html">On the Average Growth of Random Fibonacci Sequences</a>, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

%H Benoît Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, <a href="http://www.editions-belin.com/ewb_pages/f/fiche-article-quand-les-maths-se-font-discretes-8825.php">Quand les maths se font discrètes</a>, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). See p. 119. [Broken link]

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RittaudConstant.html">Rittaud Constant</a>.

%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>

%F 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

%F The largest real solution of x = sqrt(1+sqrt(-1+x)). - _Stanislav Sykora_, May 08 2016

%F From _Wolfdieter Lang_, Oct 17 2022: (Start)

%F Equals ((4*(43 + 3*sqrt(177)))^(1/3) + 16*(4*(43 + 3*sqrt(177)))^(-1/3) - 2)/6.

%F Equals ((4*(43 + 3*sqrt(177)))^(1/3) + (4*(43 - 3*sqrt(177)))^(1/3) - 2)/6.

%F Equals (4*cosh((1/3)*arccosh(43/16)) - 1)/3. (End)

%e 1.20556943040059031170202861778382342637710891959769944...

%p Digits := 80 ; fsolve( x^3-2*x^2-1,x,2.2..2.3)-1.0 ; # _R. J. Mathar_, Apr 23 2008

%t RealDigits[Root[x^3 + x^2 - x - 2, x, 1], 10, 98] // First (* _Jean-François Alcover_, Aug 06 2014 *)

%o (PARI) real(polroots(x^3+x^2-x-2)[1]) \\ _Charles R Greathouse IV_, May 28 2011

%o (PARI) polrootsreal(x^3+x^2-x-2)[1] \\ _Charles R Greathouse IV_, May 14 2014

%Y Cf. A078416, A272874. A357468.

%K easy,nonn,cons,nice

%O 1,2

%A _Jonathan Vos Post_, Apr 16 2008

%E More terms from _R. J. Mathar_, Apr 23 2008

%E More terms from _Jean-François Alcover_, Aug 06 2014