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

%I #15 Nov 09 2022 19:20:14

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

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

%U 4,0,1,0,3,9,0,1

%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 - 11/27.

%C The other (complex) roots of 2*x^3 + x^2 - x - 1 are (-1 + w1*(44 + 3*sqrt(177))^(1/3) + w2*(44 - 3*sqrt(177))^(1/3))/6 = -0.6647417704... + 0.4011272787...*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((44/49)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((44/49)*sqrt(7)))*i))/6, and its complex conjugate.

%F r = (-1 + (44 + 3*sqrt(177))^(1/3) + 7*(44 + 3*sqrt(177))^(-1/3))/6.

%F r = (-1 + (44 + 3*sqrt(177))^(1/3) + (44 - 3*sqrt(177))^(1/3))/6.

%F r = (-1 + 2*sqrt(7)*cosh((1/3)*arccosh((44/49)*sqrt(7))))/6.

%F r = (-1/6) + (2^(2/3)*11^(1/3))/3 * Hyper2F1([-1/6,1/3],[1/2],1593/1936). - _Gerry Martens_, Nov 08 2022

%e 0.82948354095849703967338757839200780762199667228138855017610774449208401039...

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

%o (PARI) (-1/6) + (2^(2/3)*11^(1/3))/3 * hypergeom([-1/6,1/3],[1/2],1593/1936) \\ _Michel Marcus_, Nov 08 2022

%Y Cf. A358182, A358184.

%K nonn,cons,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 07 2022