A357470 Decimal expansion of the real root of x^3 - x^2 - 2*x - 1.

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

Offset: 1

Author

Wolfdieter Lang, Oct 25 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 + 1/3 where r0 is the real root of y^3 - (7/3)*y - 47/27.

The other roots of x^3 - x^2 - 2*x - 1 are (2 + w1*(4*(47 + 3*sqrt(93)))^(1/3) + w2*(4*(47 - 3*sqrt(93)))^(1/3))/6 = -0.5739495178... + 0.3689894074...*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 roots of x^3 - 1.

Using hyperbolic functions these roots are (1 - sqrt(7)*(cosh((1/3)*arccosh((47/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((47/98)*sqrt(7)))*i))/3, and its complex conjugate.

Examples

			2.147899035704787354026214964930987364916766150370284279446911717889159675...
		

Crossrefs

Cf. A160389, A255524, A255249, A357471, A357472.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 - x^2 - 2*x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)

Formula

r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + 28*(4*(47 + 3*sqrt(93)))^(-1/3))/6.

r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + (4*(47 - 3*sqrt(93)))^(1/3))/6.

r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((47/98)*sqrt(7))))/3.

r = (1/3) + (188^(1/3)/3)*Hyper2F1([-1/6, 1/3], [1/2], 837/(47^2)). - Gerry Martens, Nov 04 2022