A114431 - OEIS

%I #19 Apr 15 2020 11:39:04

%S 2,4,6,7,5,0,3,8,5,7,0,5,6,5,1,7,5,7,6,3,8,1,8,8,6,7,5,5,3,5,8,7,8,6,

%T 0,7,0,3,8,2,2,5,4,4,7,5,0,6,2,3,7,2,9,8,8,4,6,4,0,1,9,7,7,4,0,5,5,0,

%U 7,5,1,9,3,5,9,1,7,7,3,3,9,7,1,5,8,1,5,9,5,1,6,3,4,9,2,3,8,6,3,5,7,5,3,9,3

%N Decimal expansion of the real solution of x^3 - x^2 - 2x - 4 = 0.

%C The solution of the equation is twice the value of an upper bound on randomly generated Fibonacci-like sequences.

%C Also, 1/log_2(x), where x is this constant, is the exponent in the exponent of the growth rate of the first Grigorchuk group. - _Andrey Zabolotskiy_, Apr 14 2020

%H Anna Erschler & Tianyi Zheng, <a href="https://doi.org/10.1007/s00222-019-00922-0">Growth of periodic Grigorchuk groups</a>, Invent. math. 219, 1069-1155 (2020).

%H Eran Makover and Jeffrey McGowan, <a href="http://dx.doi.org/10.1016/j.jnt.2006.01.002">An elementary proof that random Fibonacci sequences grow exponentially</a>, Journal of Number Theory, Volume 121, Issue 1, November 2006, Pages 40-44. (arXiv:math/0510159 [math.NT])

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Grigorchuk_group">Grigorchuk group</a>

%t RealDigits[x/.FindRoot[x^3 - x^2 - 2x == 4, {x, 2}, WorkingPrecision -> 120], 10, 120] [[1]] (* or *) RealDigits[(1 + Surd[64 - 3 * Sqrt[417], 3] + Surd[64 + 3 * Sqrt[417], 3])/3, 10, 120][[1]] (* _Harvey P. Dale_, Dec 02 2017 *)

%o (PARI) default(realprecision, 105); 1/3*(1+(64-3*sqrt(417))^(1/3)+(64+3*sqrt(417))^(1/3)) \\ _Michel Marcus_, Jun 14 2013

%Y Cf. A114427.

%K cons,nonn

%O 0,1

%A _Stefan Steinerberger_, Feb 13 2006