OFFSET
1,2
COMMENTS
This equals r0 + 1/3 where r0 is the real root of y^3 - (1/3)*y - 31/54.
The other roots of 2*x^3 - 2*x^2 - 1 are (w1*((31 + 3*sqrt(105))/4)^(1/3) + w2*((31 - 3*sqrt(105))/4)^(1/3))/3 = -0.4819115874... + 0.6028125753...*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.
Using hyperbolic functions these roots are (-cosh((1/3)*arccosh(31/4)) + sqrt(3)*sinh((1/3)*arccosh(31/4))*i)/3, and its complex conjugate.
FORMULA
r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 + 3*sqrt(105))/4)^(-1/3) + 1)/3.
r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 - 3*sqrt(105))/4)^(1/3) + 1)/3.
r = (2*cosh((1/3)*arccosh(31/4))+1)/3.
EXAMPLE
1.29715650817742437246783022983731955553805581370396822836159443088438391495...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 - 2*x^2 - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 29 2022
STATUS
approved