OFFSET
0,1
COMMENTS
This equals r0 - 1/6 where r0 is the real root of y^3 - (1/12)*y - 107/108.
The complex roots of 2*x^3 + x^2 - 2 are (w1*(107 + 6*sqrt(318))^(1/3) + w2*(107 - 6*sqrt(318))^(1/3) - 1)/6 = -0.6790471647... + 0.8392067630...*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 (-(1 + cosh((1/3)*arccosh(107))) + sqrt(3)*sinh((1/3)*arccosh(107))*i)/6, and its complex conjugate.
FORMULA
r = ((107 + 6*sqrt(318))^(1/3) + (107 + 6*sqrt(318))^(-1/3) - 1)/6
r = ((107 + 6*sqrt(318))^(1/3) + (107 - 6*sqrt(318))^(1/3) - 1)/6.
r = (2*cosh((1/3)*arccosh(107)) - 1)/6.
EXAMPLE
0.858094329496552706258725850958187515309026929286713666496137417447921455...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 + x^2 - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
RealDigits[Root[2x^3+x^2-2, 1], 10, 100][[1]] (* Harvey P. Dale, Sep 25 2023 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 29 2022
STATUS
approved