A114431 Decimal expansion of the real solution of x^3 - x^2 - 2x - 4 = 0.

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, 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, 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

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

Table of n, a(n) for n=0..104.

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

Cf. A114427.

Sequence in context: A091476 A257578 A272665 * A167689 A058184 A087777

Adjacent sequences: A114428 A114429 A114430 * A114432 A114433 A114434

Keyword

cons,nonn

Author

Stefan Steinerberger, Feb 13 2006

Status

approved