OFFSET
1,1
COMMENTS
This equals r0 + 1/3 where r0 is the real root of y^3 - (7/3)*y - 47/27.
The other roots of x^3 - x^2 - 2*x - 1 are (2 + w1*(4*(47 + 3*sqrt(93)))^(1/3) + w2*(4*(47 - 3*sqrt(93)))^(1/3))/6 = -0.5739495178... + 0.3689894074...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - sqrt(7)*(cosh((1/3)*arccosh((47/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((47/98)*sqrt(7)))*i))/3, and its complex conjugate.
FORMULA
r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + 28*(4*(47 + 3*sqrt(93)))^(-1/3))/6.
r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + (4*(47 - 3*sqrt(93)))^(1/3))/6.
r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((47/98)*sqrt(7))))/3.
r = (1/3) + (188^(1/3)/3)*Hyper2F1([-1/6, 1/3], [1/2], 837/(47^2)). - Gerry Martens, Nov 04 2022
EXAMPLE
2.147899035704787354026214964930987364916766150370284279446911717889159675...
MATHEMATICA
RealDigits[x /. FindRoot[x^3 - x^2 - 2*x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 25 2022
STATUS
approved