OFFSET
1,2
COMMENTS
The two real and two complex roots are given by the negative roots of x^4 + 2*x - 1 (see A358189).
FORMULA
r = (sqrt(2)*u + sqrt(-2*u^2 + 2*sqrt(2*u)))/(2*sqrt(u)), where u = (((3/4)*( 9 + sqrt(129)))^(1/3) + w1*((3/4)*( 9 - sqrt(129)))^(1/3))/3 = 0.4238537990..., and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3), is one of the complex roots of x^3 - 1. Alternatively u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))). A358189 uses the same u.
EXAMPLE
1.3953369944670730187931436130710553428418349124097566207933090113521308...
MAPLE
a := ((3*sqrt(3) + sqrt(43))/4)^(1/3): b := (a - 1/a)/sqrt(3):
c := (sqrt(sqrt(2*b) - b^2) + b)/(sqrt(2*b)): evalf(c, 76); # Peter Luschny, Dec 10 2022
MATHEMATICA
RealDigits[x /. FindRoot[x^4 - 2*x - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 07 2022
STATUS
approved