OFFSET
0,1
COMMENTS
This equals r0 + 1/3 where r0 is the real root of y^3 + (5/3)*y - 11/27.
The other roots of x^3 - x^2 + 2*x - 1 are (1 + w1*((11 + 3*sqrt(69))/2)^(1/3) + ((11 - 3*sqrt(69))/2)^(1/3))/3 = 0.2150798545... + 1.3071412786...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - sqrt(5)*(sinh((1/3)*arcsinh((11/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((11/50)*sqrt(5)))*i))/3, and its complex conjugate.
FORMULA
r = (2 + (4*(11 + 3*sqrt(69)))^(1/3) - 20*(4*(11 + 3*sqrt(69)))^(-1/3))/6.
r = (2 + (4*(11 + 3*sqrt(69)))^(1/3) + w1*(4*(11 - 3*sqrt(69)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex conjugate roots of x^3 - 1.
r = (1 + 2*sqrt(5)*sinh((1/3)*arcsinh((11/50)*sqrt(5))))/3.
r = (1/3) + (11/45)*Hyper2F1([1/3, 2/3],[3/2], -(11^2)/(2^2*5^3)). - Gerry Martens, Nov 04 2022
EXAMPLE
0.569840290998053265911399958119568648839797439128940220544731079656747...
MATHEMATICA
RealDigits[x /. FindRoot[x^3 - x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Oct 25 2022
STATUS
approved