OFFSET
0,1
COMMENTS
This equals r0 - 1/3 where r0 is the real root of y^3 + (5/3)*y - 43/27.
The other roots of x^3 - x^2 + 2*x - 1 are (-1 + w1*((43 + 9*sqrt(29))/2)^(1/3) + ((43 - 9*sqrt(29))/2)^(1/3))/3 = -0.6963233908... + 1.4359498641...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are -(1 + sqrt(5)*(sinh((1/3)*arcsinh( (43/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((43/50)*sqrt(5)))*i))/3, and its complex conjugate.
FORMULA
r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) - 20*(4*(43 + 9*sqrt(29)))^(-1/3))/6.
r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) + w1*(4*(43 - 9*sqrt(29)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i) is one of the complex conjugate roots of x^3 - 1.
r = (-1 + 2*sqrt(5)*sinh((1/3)*arcsinh((43/50)*sqrt(5))))/3.
r = (-5*u^(1/3) + u^(-1/3) - 1)/3 where u = 2/(43 + 9*sqrt(29)). - Peter Luschny, Nov 01 2022
r = (-1/3) + (43/45) * Hyper2F1([1/3, 2/3], [3/2], -43^2/(5*10^2)). - Gerry Martens, Nov 04 2022
EXAMPLE
0.3926467817026408117648795948843412507037649685934825897311396498445171668...
MAPLE
Digits:=100: u := 2/(43 + 9*sqrt(29)): (-5*u^(1/3) + u^(-1/3) - 1)/3:
evalf(%*10^78): ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 01 2022
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