A357107 Decimal expansion of the real root of 2*x^3 - x - 2.

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

Offset

1,3

Comments

The complex roots are (w1*(4 + (2/9)*sqrt(318))^(1/3) + w2*(4 - (2/9)*sqrt(318))^(1/3))/2 = -0.5826865215... + 0.7201185646...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are (1/6)*sqrt(6)*(-cosh((1/3)*arccosh(3*sqrt(6))) + sqrt(3)*sinh((1/3)*arccosh(3*sqrt(6)))*i), and its complex conjugate.

Links

Table of n, a(n) for n=1..78.

Formula

r = ((108 + 6*sqrt(318))^(1/3) + 6*(108 + 6*sqrt(318))^(-1/3))/6.

r = (3*(4 + (2/9)*sqrt(318))^(1/3) + 2*(4 + (2/9)*sqrt(318))^(-1/3))/6.

r = ((4 + (2/9)*sqrt(318))^(1/3) + (4 - (2/9)*sqrt(318))^(1/3))/2.

r = (1/3)*sqrt(6)*cosh((1/3)*arccosh(3*sqrt(6))).

Example

1.165373043062414716956358434517798082542887318820048613344266311648448471...

Mathematica

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

Crossrefs

Cf. A357108.

Sequence in context: A143304 A021157 A242494 * A193211 A195713 A306712

Adjacent sequences: A357104 A357105 A357106 * A357108 A357109 A357110

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 29 2022

Status

approved