OFFSET
0,1
COMMENTS
This equals r0 - 1/3 where r0 is the real root of y^3 + (2/3)*y - 61/27.
The other roots of x^3 + x^2 + x - 2 are (w1*(4*(61 + 3*sqrt(417)))^(1/3) + (4*(61 - 3*sqrt(417)))^(1/3) - 2)/6 = -0.9052678568... + 1.2837421720...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are -(1/3)*(1 + sqrt(2)*(sinh((1/3)*arcsinh((61/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((61/8)*sqrt(2)))*i)), and its complex conjugate.
FORMULA
r = ((4*(61 + 3*sqrt(417)))^(1/3) - 8*(4*(61 + 3*sqrt(417)))^(-1/3) - 2)/6.
r = ((4*(61 + 3*sqrt(417)))^(1/3) + w1*(4*(61 - 3*sqrt(417)))^(1/3) - 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (-1 + 2*sqrt(2)*sinh((1/3)*arcsinh((61/8)*sqrt(2))))/3.
EXAMPLE
0.8105357137661367740212514143256682141072614900005302474430976745094594...
MATHEMATICA
RealDigits[x /. FindRoot[x^3 + x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 17 2022
STATUS
approved