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

%I #14 Nov 09 2022 19:20:05

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

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

%U 3,7,5,9,6,7,9,5

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

%C This equals r0 + 1/6 where r0 is the real root of y^3 - (7/12)*y - 16/27.

%C The other roots of 2*x^3 - x^2 - x - 1 are (1 + w1*(64 + 3*sqrt(417))^(1/3) + w2*(64 - 3*sqrt(417))^(1/3))/6 = -0.3668759642... + 0.5202594388...*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 (1 - sqrt(7)*( cosh((1/3)*arccosh((64/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((64/49)*sqrt(7)))*i))/6, and its complex conjugate.

%F r = (1 + (64 + 3*sqrt(417))^(1/3) + 7*(64 + 3*sqrt(417))^(-1/3))/6.

%F r = (1 + (64 + 3*sqrt(417))^(1/3) + (64 - 3*sqrt(417))^(1/3))/6.

%F r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((64/49)*sqrt(7))))/6.

%F r = (1/6) + (4/3)*Hyper2F1([-1/6,1/3],[1/2],3753/4096). - _Gerry Martens_, Nov 08 2022

%e 1.23375192852825878819094337767939303519112723753118649423200988702753759...

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

%Y Cf. A358181, A358183.

%K nonn,cons,easy

%O 1,2

%A _Wolfdieter Lang_, Nov 07 2022