A358181 - OEIS

%I #15 Nov 09 2022 19:19:50

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

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

%U 9,7,9,8,3,1,3,1

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

%C This equals r0 + 2/3 where r0 is the real root of y^3 - (7/3)*y - 61/27.

%C The other roots of x^3 - 2*x^2 - x - 1 are (2 + w1*((61 + 9*sqrt(29))/2)^(1/3) + w2*((61 - 9*sqrt(29))/2)^(1/3))/3 = -0.2734091384... + 0.5638210928...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.

%C Using hyperbolic functions these roots are (2 - sqrt(7)*(cosh((1/3)*arccosh((61/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((61/98)*sqrt(7)))*i))/3, and its complex conjugate.

%F r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + 7*((61 + 9*sqrt(29))/2)^(-1/3))/3.

%F r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + ((61 - 9*sqrt(29))/2)^(1/3))/3.

%F r = 2*(1 + sqrt(7)*cosh((1/3)*arccosh((61/98)*sqrt(7))))/3.

%F r = (2/3) +(2^(2/3)*61^(1/3))/3*Hyper2F1([-1/6,1/3],[1/2],2349/3721). - _Gerry Martens_, Nov 08 2022

%e 2.5468182768840820791359975088097915288112703374520061295514745747111979831...

%t RealDigits[x /. FindRoot[x^3 - 2*x^2 - x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Nov 08 2022 *)

%Y Cf. A088559, A231187, 1 - A160389, A255240.

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Nov 07 2022