OFFSET
0,1
COMMENTS
The solution of the equation is twice the value of an upper bound on randomly generated Fibonacci-like sequences.
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
LINKS
Anna Erschler & Tianyi Zheng, Growth of periodic Grigorchuk groups, Invent. math. 219, 1069-1155 (2020).
Eran Makover and Jeffrey McGowan, An elementary proof that random Fibonacci sequences grow exponentially, Journal of Number Theory, Volume 121, Issue 1, November 2006, Pages 40-44. (arXiv:math/0510159 [math.NT])
Wikipedia, Grigorchuk group
MATHEMATICA
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 *)
PROG
(PARI) default(realprecision, 105); 1/3*(1+(64-3*sqrt(417))^(1/3)+(64+3*sqrt(417))^(1/3)) \\ Michel Marcus, Jun 14 2013
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Stefan Steinerberger, Feb 13 2006
STATUS
approved