OFFSET
0,1
COMMENTS
This equals r0 - 1/9 where r0 is the real root of y^3 - (1/27)*y - 241/729.
The other (complex) roots of 3*x^3 + x^2 - 1 are (w1*(4*(241 + 9*sqrt(717)))^(1/3) + w2*(4*(241 - 9*sqrt(717)))^(1/3) - 2)/18 = -0.4657634157... + 0.5833504388...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1 + cosh((1/3)*arccosh(241/2)) - sqrt(3)*sinh((1/3)*arccosh(241/2))*i)/9 and its complex conjugate.
FORMULA
r = ((4*(241 + 9*sqrt(717)))^(1/3) + 4*(4*(241 + 9*sqrt(717)))^(-1/3) - 2)/18.
r = ((4*(241 + 9*sqrt(717)))^(1/3) + (4*(241 - 9*sqrt(717)))^(1/3) - 2)/18.
r = (2*cosh((1/3)*arccosh(241/2)) - 1)/9.
EXAMPLE
0.59819349811085533042783790621004944673398424715056106803235989051103...
MATHEMATICA
RealDigits[x /. FindRoot[3*x^3 + x^2 - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 07 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 30 2022
STATUS
approved