A357103 Decimal expansion of the real root of x^3 - 3*x - 3.

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

Offset

1,1

Comments

This equals the real root of x^3 - 3*x^2 - 1 if 1 is added.

The other two roots of x^3 - 3*x - 3 are w1*phi^(2/3) + w2*phi^(-2/3) = -1.0519017013... + 0.5652358516...*i, and its complex conjugate, where phi = A001622, and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these complex roots are cosh((1/3)*arccosh(3/2)) + sqrt(3)*sinh((1/3)*arccosh(3/2))*i, and its complex conjugate.

Links

Table of n, a(n) for n=1..76.

Formula

r = (1 + phi)^(1/3) + (1 + phi)^(-1/3), with the golden section phi = A001622.

r = (1 + phi)^(1/3) + (2 - phi)^(1/3).

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

Example

2.103803402735536533164947332828928092419417083230268513734743062120983716...

Maple

h := ((3 + sqrt(5))/2)^(1/3): evalf(h + 1/h, 90); # Peter Luschny, Sep 24 2022

Mathematica

RealDigits[Plus @@ Surd[GoldenRatio + 1, {3, -3}], 10, 100][[1]] (* Amiram Eldar, Sep 21 2022 *)

Prog

(PARI) 2*cosh((1/3)*acosh(3/2)) \\ Michel Marcus, Sep 23 2022

Crossrefs

Cf. A001622, A357104.

Sequence in context: A230360 A244117 A263426 * A278882 A153007 A090683

Adjacent sequences: A357100 A357101 A357102 * A357104 A357105 A357106

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 20 2022

Status

approved