%I #13 Jan 11 2023 15:59:25
%S 7,1,6,6,7,2,7,4,9,2,8,2,2,8,6,6,3,8,4,2,4,7,3,9,3,0,1,4,3,2,5,5,6,1,
%T 8,3,9,2,1,5,5,1,3,7,6,0,2,9,8,6,1,6,4,6,6,7,8,9,4,5,6,8,0,2,4,2,1,4,
%U 7,4,9,0,0,7,3,3,8,7
%N Decimal expansion of the positive real root r of x^4 + 2*x^3 - 1.
%C The roots of x^4 + 2*x^3 - 1 are X = Y - 1/2, where Y are the roots of y^4 - (3/2)*y^2 + y - 19/16.
%C The other real root is -A358188, and the complex roots are -0.3048767044... + 0.7545291731...*i and its complex conjugate. The formula for the first complex root is obtained by replacing 3 and the two 4s with their negative values in the formula for r given below, while using the unchanged u.
%F r = (1/2)*((-3 - 4*u + sqrt(9 - 16*u^2 + 4*sqrt(8*u + 6)))/sqrt(8*u + 6) - 1), with u = (((3/4)*(-9 + sqrt(129)))^(1/3) + w1*((3/4)*(-9 - sqrt(129)))^(1/3) - 3/4)/3 = -0.06738537990..., where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1. Or with hyperbolic functions u = -(2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))) - 1/4.
%e 0.716672749282286638424739301432556183921551376029861646678945680242147...
%t RealDigits[x /. FindRoot[x^4 + 2*x^3 - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* _Amiram Eldar_, Dec 07 2022 *)
%Y Cf. A358188.
%K nonn,cons,easy
%O 0,1
%A _Wolfdieter Lang_, Dec 06 2022