%I #10 Dec 15 2022 17:01:34

%S 8,1,0,5,3,5,7,1,3,7,6,6,1,3,6,7,7,4,0,2,1,2,5,1,4,1,4,3,2,5,6,6,8,2,

%T 1,4,1,0,7,2,6,1,4,9,0,0,0,0,5,3,0,2,4,7,4,4,3,0,9,7,6,7,4,5,0,9,4,5,

%U 9,4,0,8,7,4,7,2

%N Decimal expansion of the real root of x^3 + x^2 + x - 2.

%C This equals r0 - 1/3 where r0 is the real root of y^3 + (2/3)*y - 61/27.

%C The other roots of x^3 + x^2 + x - 2 are (w1*(4*(61 + 3*sqrt(417)))^(1/3) + (4*(61 - 3*sqrt(417)))^(1/3) - 2)/6 = -0.9052678568... + 1.2837421720...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.

%C Using hyperbolic functions these roots are -(1/3)*(1 + sqrt(2)*(sinh((1/3)*arcsinh((61/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((61/8)*sqrt(2)))*i)), and its complex conjugate.

%F r = ((4*(61 + 3*sqrt(417)))^(1/3) - 8*(4*(61 + 3*sqrt(417)))^(-1/3) - 2)/6.

%F r = ((4*(61 + 3*sqrt(417)))^(1/3) + w1*(4*(61 - 3*sqrt(417)))^(1/3) - 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.

%F r = (-1 + 2*sqrt(2)*sinh((1/3)*arcsinh((61/8)*sqrt(2))))/3.

%e 0.8105357137661367740212514143256682141072614900005302474430976745094594...

%t RealDigits[x /. FindRoot[x^3 + x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 18 2022 *)

%Y Cf. A137421.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Oct 17 2022