A357465 - OEIS

%I #9 Nov 09 2022 05:02:08

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

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

%U 8,0,9,8,6,8,1,2

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

%C This equals r0 + 1/9 where r0 is the real root of y^3 - (1/27)*y - 245/729.

%C The other roots of 3*x^3 - x^2 - 1 are (w1*(4*(245 + 9*sqrt(741)))^(1/3) + w2*(4*(245 - 9*sqrt(741)))^(1/3) + 2)/18  = -0.2453946438... + 0.5867299404...*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(245/2)) + sqrt(3)*sinh((1/3)*arccosh(245/2))*i)/9 and its complex conjugate.

%F r = ((4*(245 + 9*sqrt(741)))^(1/3) + 4*(4*(245 + 9*sqrt(741)))^(-1/3) + 2)/18.

%F r = ((4*(245 + 9*sqrt(741)))^(1/3) + (4*(245 - 9*sqrt(741)))^(1/3) + 2)/18.

%F r = (1 + 2*cosh((1/3)*arccosh(245/2)))/9.

%e 0.824122621109132966312278979870282562643326414370638728916043765420978098...

%t RealDigits[x /. FindRoot[3*x^3 - x^2 - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 07 2022 *)

%Y Cf. A357464.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Sep 30 2022