A358187 Decimal expansion of the positive real root r of x^4 + 2*x^3 - 1.

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

Offset

0,1

Comments

The roots of x^4 + 2*x^3 - 1 are X = Y - 1/2, where Y are the roots of y^4 - (3/2)*y^2 + y - 19/16.

The other real root is -A358188, and the complex roots are -0.3048767044... + 0.7545291731...*i and its complex conjugate. The formula for the first complex root is obtained by replacing 3 and the two 4s with their negative values in the formula for r given below, while using the unchanged u.

Links

Table of n, a(n) for n=0..77.

Formula

r = (1/2)*((-3 - 4*u + 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

0.716672749282286638424739301432556183921551376029861646678945680242147...

Mathematica

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

Crossrefs

Cf. A358188.

Sequence in context: A092615 A199279 A086309 * A255888 A060625 A145423

Adjacent sequences: A358184 A358185 A358186 * A358188 A358189 A358190

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Dec 06 2022

Status

approved