A358181 Decimal expansion of the real root of x^3 - 2*x^2 - x - 1.

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

Offset: 1

Author

Wolfdieter Lang, Nov 07 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 + 2/3 where r0 is the real root of y^3 - (7/3)*y - 61/27.

The other roots of x^3 - 2*x^2 - x - 1 are (2 + w1*((61 + 9*sqrt(29))/2)^(1/3) + w2*((61 - 9*sqrt(29))/2)^(1/3))/3 = -0.2734091384... + 0.5638210928...*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.

Using hyperbolic functions these roots are (2 - sqrt(7)*(cosh((1/3)*arccosh((61/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((61/98)*sqrt(7)))*i))/3, and its complex conjugate.

Examples

			2.5468182768840820791359975088097915288112703374520061295514745747111979831...
		

Crossrefs

Cf. A088559, A231187, 1 - A160389, A255240.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 - 2*x^2 - x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[x^3-2x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Mar 30 2025 *)

Formula

r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + 7*((61 + 9*sqrt(29))/2)^(-1/3))/3.

r = (2 + ((61 + 9*sqrt(29))/2)^(1/3) + ((61 - 9*sqrt(29))/2)^(1/3))/3.

r = 2*(1 + sqrt(7)*cosh((1/3)*arccosh((61/98)*sqrt(7))))/3.

r = (2/3) +(2^(2/3)*61^(1/3))/3*Hyper2F1([-1/6,1/3],[1/2],2349/3721). - Gerry Martens, Nov 08 2022