A358190 Decimal expansion of the positive real root r of x^4 - 2*x - 1.

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

Offset

1,2

Comments

The two real and two complex roots are given by the negative roots of x^4 + 2*x - 1 (see A358189).

Links

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

Formula

r = (sqrt(2)*u + sqrt(-2*u^2 + 2*sqrt(2*u)))/(2*sqrt(u)), where u = (((3/4)*( 9 + sqrt(129)))^(1/3) + w1*((3/4)*( 9 - sqrt(129)))^(1/3))/3 = 0.4238537990..., and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3), is one of the complex roots of x^3 - 1. Alternatively u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))). A358189 uses the same u.

Example

1.3953369944670730187931436130710553428418349124097566207933090113521308...

Maple

a := ((3*sqrt(3) + sqrt(43))/4)^(1/3): b := (a - 1/a)/sqrt(3):
c := (sqrt(sqrt(2*b) - b^2) + b)/(sqrt(2*b)): evalf(c, 76); # Peter Luschny, Dec 10 2022

Mathematica

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

Crossrefs

Cf. A358189.

Sequence in context: A016675 A265272 A021257 * A282529 A070341 A245719

Adjacent sequences: A358187 A358188 A358189 * A358191 A358192 A358193

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Dec 07 2022

Status

approved