A357464 Decimal expansion of the real root of 3*x^3 + x^2 - 1.

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

Offset

0,1

Comments

This equals r0 - 1/9 where r0 is the real root of y^3 - (1/27)*y - 241/729.

The other (complex) roots of 3*x^3 + x^2 - 1 are (w1*(4*(241 + 9*sqrt(717)))^(1/3) + w2*(4*(241 - 9*sqrt(717)))^(1/3) - 2)/18 = -0.4657634157... + 0.5833504388...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) 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(241/2)) - sqrt(3)*sinh((1/3)*arccosh(241/2))*i)/9 and its complex conjugate.

Links

Table of n, a(n) for n=0..74.

Formula

r = ((4*(241 + 9*sqrt(717)))^(1/3) + 4*(4*(241 + 9*sqrt(717)))^(-1/3) - 2)/18.

r = ((4*(241 + 9*sqrt(717)))^(1/3) + (4*(241 - 9*sqrt(717)))^(1/3) - 2)/18.

r = (2*cosh((1/3)*arccosh(241/2)) - 1)/9.

Example

0.59819349811085533042783790621004944673398424715056106803235989051103...

Mathematica

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

Crossrefs

Cf. A357465.

Sequence in context: A255247 A366841 A153610 * A249385 A347216 A247747

Adjacent sequences: A357461 A357462 A357463 * A357465 A357466 A357467

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 30 2022

Status

approved