OFFSET
1,2
COMMENTS
This equals r0 + 1/6 where r0 is the real root of y^3 - (7/12)*y - 16/27.
The other roots of 2*x^3 - x^2 - x - 1 are (1 + w1*(64 + 3*sqrt(417))^(1/3) + w2*(64 - 3*sqrt(417))^(1/3))/6 = -0.3668759642... + 0.5202594388...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - sqrt(7)*( cosh((1/3)*arccosh((64/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((64/49)*sqrt(7)))*i))/6, and its complex conjugate.
FORMULA
r = (1 + (64 + 3*sqrt(417))^(1/3) + 7*(64 + 3*sqrt(417))^(-1/3))/6.
r = (1 + (64 + 3*sqrt(417))^(1/3) + (64 - 3*sqrt(417))^(1/3))/6.
r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((64/49)*sqrt(7))))/6.
r = (1/6) + (4/3)*Hyper2F1([-1/6,1/3],[1/2],3753/4096). - Gerry Martens, Nov 08 2022
EXAMPLE
1.23375192852825878819094337767939303519112723753118649423200988702753759...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 - x^2 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 07 2022
STATUS
approved