A357109 Decimal expansion of the real root of 2*x^3 - 2*x^2 - 1.

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

Offset: 1

Author

Wolfdieter Lang, Sep 29 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 + 1/3 where r0 is the real root of y^3 - (1/3)*y - 31/54.

The other roots of 2*x^3 - 2*x^2 - 1 are (w1*((31 + 3*sqrt(105))/4)^(1/3) + w2*((31 - 3*sqrt(105))/4)^(1/3))/3 = -0.4819115874... + 0.6028125753...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are (-cosh((1/3)*arccosh(31/4)) + sqrt(3)*sinh((1/3)*arccosh(31/4))*i)/3, and its complex conjugate.

Examples

			1.29715650817742437246783022983731955553805581370396822836159443088438391495...
		

Links

• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A273065.

Programs

Mathematica

RealDigits[x /. FindRoot[2*x^3 - 2*x^2 - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)

PARI

polrootsreal(2*x^3 - 2*x^2 - 1)[1] \\ Charles R Greathouse IV, Feb 11 2025

Formula

r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 + 3*sqrt(105))/4)^(-1/3) + 1)/3.

r = (((31 + 3*sqrt(105))/4)^(1/3) + ((31 - 3*sqrt(105))/4)^(1/3) + 1)/3.

r = (2*cosh((1/3)*arccosh(31/4))+1)/3.