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Decimal expansion of the real root of 2*x^3 + x - 2.
1

%I #6 Oct 13 2022 13:04:22

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

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

%U 9,5,1,8,1,0,9,8,0,8

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

%C The other (complex) roots are w1*(1/2 + (1/36)*sqrt(330))^(1/3) + (1/2 - (1/36)*sqrt(330))^(1/3) = -0.4175611742... + 1.0114702183....*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) is one of 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(330))^(1/3) - 6*(108 + 6*sqrt(330))^(-1/3))/6.

%F r = (6*(1/2 + (1/36)*sqrt(330))^(1/3) - (1/2 + (1/36)*sqrt(330))^(-1/3))/6.

%F r = (1/2 + (1/36)*sqrt(330))^(1/3) + w1*(1/2 - (1/36)*sqrt(330))^(1/3), where w1 = (-1 = sqrt(3)*i)/2 is one of the complex roots of x^3 - 1.

%F r= (1/3)*sqrt(6)*sinh((1/3)*arcsinh(3*sqrt(6))).

%e 0.835122348481366514291620038596702299165411487780433601936279731538589...

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

%Y Cf. A357107.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Sep 29 2022