A357107 - OEIS

%I #6 Oct 13 2022 13:04:14

%S 1,1,6,5,3,7,3,0,4,3,0,6,2,4,1,4,7,1,6,9,5,6,3,5,8,4,3,4,5,1,7,7,9,8,

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

%U 4,8,4,7,1,4,0,1,1,5

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

%C The complex roots are (w1*(4 + (2/9)*sqrt(318))^(1/3) + w2*(4 - (2/9)*sqrt(318))^(1/3))/2  = -0.5826865215... + 0.7201185646...*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 (1/6)*sqrt(6)*(-cosh((1/3)*arccosh(3*sqrt(6))) + sqrt(3)*sinh((1/3)*arccosh(3*sqrt(6)))*i), and its complex conjugate.

%F r = ((108 + 6*sqrt(318))^(1/3) + 6*(108 + 6*sqrt(318))^(-1/3))/6.

%F r = (3*(4 + (2/9)*sqrt(318))^(1/3) + 2*(4 + (2/9)*sqrt(318))^(-1/3))/6.

%F r = ((4 + (2/9)*sqrt(318))^(1/3) + (4 - (2/9)*sqrt(318))^(1/3))/2.

%F r = (1/3)*sqrt(6)*cosh((1/3)*arccosh(3*sqrt(6))).

%e 1.165373043062414716956358434517798082542887318820048613344266311648448471...

%t RealDigits[x /. FindRoot[2*x^3 - x - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* _Amiram Eldar_, Sep 29 2022 *)

%Y Cf. A357108.

%K nonn,cons,easy

%O 1,3

%A _Wolfdieter Lang_, Sep 29 2022