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%I #10 Dec 11 2022 10:29:50
%S 2,1,0,6,9,1,9,3,4,0,3,7,6,2,1,7,2,1,7,0,9,7,1,0,6,1,2,9,5,3,7,9,7,3,
%T 0,4,6,6,2,9,2,7,6,5,4,4,0,9,2,8,1,4,9,3,8,3,6,7,3,5,4,6,6,4,4,1,4,2,
%U 2,4,2,7,2,9,4,2,3,7
%N Decimal expansion of the positive real root r of x^4 - 2*x^3 - 1.
%C The two real and two complex roots X of x^4 - 2*x^3 - 1 are the negative of the roots of x^4 + 2*x^3 - 1 (see A358187 for the formulas), and X = Y + 1/2, where Y are the roots of y^4 - (3/2)*y^2 - y - 19/16.
%F r = (1/2)*((4*u + 3 + 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 2.10691934037621721709710612953797304662927654409281493836735466441422427...
%t RealDigits[x /. FindRoot[x^4 - 2*x^3 - 1, {x, 2}, WorkingPrecision -> 120], 10, 120][[1]] (* _Amiram Eldar_, Dec 07 2022 *)
%Y Cf. A358187.
%K nonn,cons,easy
%O 1,1
%A _Wolfdieter Lang_, Dec 06 2022