A358188 Decimal expansion of the positive real root r of x^4 - 2*x^3 - 1.

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, 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, 2, 4, 2, 7, 2, 9, 4, 2, 3, 7

Offset

1,1

Comments

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.

Links

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

Formula

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.

Example

2.10691934037621721709710612953797304662927654409281493836735466441422427...

Mathematica

RealDigits[x /. FindRoot[x^4 - 2*x^3 - 1, {x, 2}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)

Crossrefs

Cf. A358187.

Sequence in context: A288874 A356545 A187555 * A117651 A373426 A268728

Adjacent sequences: A358185 A358186 A358187 * A358189 A358190 A358191

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Dec 06 2022

Status

approved