A357465 Decimal expansion of the real root of 3*x^3 - x^2 - 1.

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

Offset

0,1

Comments

This equals r0 + 1/9 where r0 is the real root of y^3 - (1/27)*y - 245/729.

The other roots of 3*x^3 - x^2 - 1 are (w1*(4*(245 + 9*sqrt(741)))^(1/3) + w2*(4*(245 - 9*sqrt(741)))^(1/3) + 2)/18 = -0.2453946438... + 0.5867299404...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are (1 - cosh((1/3)*arccosh(245/2)) + sqrt(3)*sinh((1/3)*arccosh(245/2))*i)/9 and its complex conjugate.

Links

Table of n, a(n) for n=0..75.

Formula

r = ((4*(245 + 9*sqrt(741)))^(1/3) + 4*(4*(245 + 9*sqrt(741)))^(-1/3) + 2)/18.

r = ((4*(245 + 9*sqrt(741)))^(1/3) + (4*(245 - 9*sqrt(741)))^(1/3) + 2)/18.

r = (1 + 2*cosh((1/3)*arccosh(245/2)))/9.

Example

0.824122621109132966312278979870282562643326414370638728916043765420978098...

Mathematica

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

Crossrefs

Cf. A357464.

Sequence in context: A298571 A096257 A333206 * A125578 A083729 A019775

Adjacent sequences: A357462 A357463 A357464 * A357466 A357467 A357468

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 30 2022

Status

approved