This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137421 #59 Oct 27 2023 10:25:27 %S A137421 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 A137421 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 A137421 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 A137421 Decimal expansion of growth constant in random Fibonacci sequence. %C A137421 Real zero of x^3 + x^2 - x - 2. - _Charles R Greathouse IV_, May 28 2011 %C A137421 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 A137421 From _Wolfdieter Lang_, Oct 17 2022: (Start) %C A137421 This equals r0 - 1/3 where r0 is the real root of y^3 - (4/3)*y - 43/27. %C A137421 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 A137421 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 A137421 (End) %H A137421 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 A137421 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 A137421 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 A137421 <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a> %F A137421 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 A137421 The largest real solution of x = sqrt(1+sqrt(-1+x)). - _Stanislav Sykora_, May 08 2016 %F A137421 From _Wolfdieter Lang_, Oct 17 2022: (Start) %F A137421 Equals ((4*(43 + 3*sqrt(177)))^(1/3) + 16*(4*(43 + 3*sqrt(177)))^(-1/3) - 2)/6. %F A137421 Equals ((4*(43 + 3*sqrt(177)))^(1/3) + (4*(43 - 3*sqrt(177)))^(1/3) - 2)/6. %F A137421 Equals (4*cosh((1/3)*arccosh(43/16)) - 1)/3. (End) %e A137421 1.20556943040059031170202861778382342637710891959769944... %p A137421 Digits := 80 ; fsolve( x^3-2*x^2-1,x,2.2..2.3)-1.0 ; # _R. J. Mathar_, Apr 23 2008 %t A137421 RealDigits[Root[x^3 + x^2 - x - 2, x, 1], 10, 98] // First (* _Jean-François Alcover_, Aug 06 2014 *) %o A137421 (PARI) real(polroots(x^3+x^2-x-2)[1]) \\ _Charles R Greathouse IV_, May 28 2011 %o A137421 (PARI) polrootsreal(x^3+x^2-x-2)[1] \\ _Charles R Greathouse IV_, May 14 2014 %Y A137421 Cf. A078416, A272874. A357468. %K A137421 easy,nonn,cons,nice %O A137421 1,2 %A A137421 _Jonathan Vos Post_, Apr 16 2008 %E A137421 More terms from _R. J. Mathar_, Apr 23 2008 %E A137421 More terms from _Jean-François Alcover_, Aug 06 2014