OFFSET
1,1
COMMENTS
This equals the real root of x^3 - 3*x^2 - 1 if 1 is added.
The other two roots of x^3 - 3*x - 3 are w1*phi^(2/3) + w2*phi^(-2/3) = -1.0519017013... + 0.5652358516...*i, and its complex conjugate, where phi = A001622, and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these complex roots are cosh((1/3)*arccosh(3/2)) + sqrt(3)*sinh((1/3)*arccosh(3/2))*i, and its complex conjugate.
FORMULA
r = (1 + phi)^(1/3) + (1 + phi)^(-1/3), with the golden section phi = A001622.
r = (1 + phi)^(1/3) + (2 - phi)^(1/3).
r = 2*cosh((1/3)*arccosh(3/2)).
EXAMPLE
2.103803402735536533164947332828928092419417083230268513734743062120983716...
MAPLE
h := ((3 + sqrt(5))/2)^(1/3): evalf(h + 1/h, 90); # Peter Luschny, Sep 24 2022
MATHEMATICA
RealDigits[Plus @@ Surd[GoldenRatio + 1, {3, -3}], 10, 100][[1]] (* Amiram Eldar, Sep 21 2022 *)
PROG
(PARI) 2*cosh((1/3)*acosh(3/2)) \\ Michel Marcus, Sep 23 2022
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 20 2022
STATUS
approved