%I #9 Nov 09 2022 04:52:09
%S 1,1,9,7,4,2,9,3,3,6,9,3,3,0,3,2,9,7,1,5,5,9,3,0,0,2,8,7,7,9,4,7,2,1,
%T 7,3,7,1,4,0,7,5,6,0,8,6,3,2,3,9,5,8,6,4,9,3,8,1,7,5,1,3,5,8,8,5,3,3,
%U 1,5,7,0,7,3,5,6,0,9
%N Decimal expansion of the real root of 2*x^3 - x^2 - 2.
%C This equals r0 + 1/6 where r0 is the real root of y^3 - (1/12)*y - 109/108.
%C The complex roots of 2*x^3 - x^2 - 2 are (w1*(109 + 6*sqrt(330))^(1/3) + w2*(109 - 6*sqrt(330))^(1/3) + 1)/6 = -0.3487146684... + 0.8447013842...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
%C Using hyperbolic functions these roots are (-cosh((1/3)*arccosh(109)) + sqrt(3)*sinh((1/3)*arccosh(109))*i)/6, and its complex conjugate.
%F r = ((109 + 6*sqrt(330))^(1/3) + (109 + 6*sqrt(330))^(-1/3) + 1)/6.
%F r = ((109 + 6*sqrt(330))^(1/3) + (109 - 6*sqrt(330))^(1/3) + 1)/6.
%F r = (2*cosh((1/3)*arccosh(109)) + 1)/6.
%e 1.197429336933032971559300287794721737140756086323958649381751358853315707...
%t RealDigits[x /. FindRoot[2*x^3 - x^2 - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* _Amiram Eldar_, Sep 29 2022 *)
%Y Cf. A357106.
%K nonn,cons,easy
%O 1,3
%A _Wolfdieter Lang_, Sep 29 2022