A357106 - OEIS

%I #10 Sep 25 2023 13:10:10

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

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

%U 1,4,5,5,3,0,3,3,4,8

%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 - 107/108.

%C The complex roots of 2*x^3 + x^2 - 2 are (w1*(107 + 6*sqrt(318))^(1/3) + w2*(107 - 6*sqrt(318))^(1/3) - 1)/6 = -0.6790471647... + 0.8392067630...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and  w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 -1.

%C Using hyperbolic functions these roots are (-(1 + cosh((1/3)*arccosh(107)))  + sqrt(3)*sinh((1/3)*arccosh(107))*i)/6, and its complex conjugate.

%F r = ((107 + 6*sqrt(318))^(1/3) + (107 + 6*sqrt(318))^(-1/3)  - 1)/6

%F r = ((107 + 6*sqrt(318))^(1/3) + (107 - 6*sqrt(318))^(1/3)  - 1)/6.

%F r = (2*cosh((1/3)*arccosh(107)) - 1)/6.

%e 0.858094329496552706258725850958187515309026929286713666496137417447921455...

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

%t RealDigits[Root[2x^3+x^2-2,1],10,100][[1]] (* _Harvey P. Dale_, Sep 25 2023 *)

%Y Cf. A357105.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Sep 29 2022