A357466 - OEIS

%I #10 Dec 29 2022 06:23:42

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

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

%U 2,5,5,3,0,4,5,7,2,4,8,6,5,8,6,9,6

%N Decimal expansion of the real root of 3*x^3 - x - 1.

%C The other (complex) roots are (w1*(4*(9 + sqrt(77)))^(1/3) + w2*(4*(9 - sqrt(77)))^(1/3))/6 = -0.4256915364... + 0.458591887...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3))/2 are the complex roots of x^3 - 1.

%C Using hyperbolic functions these roots are (-cosh((1/3)*arccosh(9/2)) + sqrt(3)*sinh((1/3)*arccosh(9/2))*i)/3, and its complex conjugate.

%F r = ((4*(9 + sqrt(77)))^(1/3) + 4*(4*(9 + sqrt(77)))^(-1/3))/6.

%F r = ((4*(9 + sqrt(77)))^(1/3) + (4*(9 - sqrt(77)))^(1/3))/6.

%F r = (2/3)*cosh((1/3)*arccosh(9/2)).

%e 0.851383072866924393493940112187859385096149923938041965059002396279722...

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

%Y Cf. A357465, A357467.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Oct 17 2022