%I #17 Nov 14 2022 05:58:50
%S 2,1,4,7,8,9,9,0,3,5,7,0,4,7,8,7,3,5,4,0,2,6,2,1,4,9,6,4,9,3,0,9,8,7,
%T 3,6,4,9,1,6,7,6,6,1,5,0,3,7,0,2,8,4,2,7,9,4,4,6,9,1,1,7,1,7,8,8,9,1,
%U 5,9,6,7,5,3,7,2,0,1
%N Decimal expansion of the real root of x^3 - x^2 - 2*x - 1.
%C This equals r0 + 1/3 where r0 is the real root of y^3 - (7/3)*y - 47/27.
%C The other roots of x^3 - x^2 - 2*x - 1 are (2 + w1*(4*(47 + 3*sqrt(93)))^(1/3) + w2*(4*(47 - 3*sqrt(93)))^(1/3))/6 = -0.5739495178... + 0.3689894074...*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 roots of x^3 - 1.
%C Using hyperbolic functions these roots are (1 - sqrt(7)*(cosh((1/3)*arccosh((47/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((47/98)*sqrt(7)))*i))/3, and its complex conjugate.
%F r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + 28*(4*(47 + 3*sqrt(93)))^(-1/3))/6.
%F r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + (4*(47 - 3*sqrt(93)))^(1/3))/6.
%F r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((47/98)*sqrt(7))))/3.
%F r = (1/3) + (188^(1/3)/3)*Hyper2F1([-1/6, 1/3], [1/2], 837/(47^2)). - _Gerry Martens_, Nov 04 2022
%e 2.147899035704787354026214964930987364916766150370284279446911717889159675...
%t RealDigits[x /. FindRoot[x^3 - x^2 - 2*x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 26 2022 *)
%Y Cf. A160389, A255524, A255249, A357471, A357472.
%K nonn,cons,easy
%O 1,1
%A _Wolfdieter Lang_, Oct 25 2022