A357466 Decimal expansion of the real root of 3*x^3 - x - 1.

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

Offset

0,1

Comments

The other (complex) roots are (w1*(4*(9 + sqrt(77)))^(1/3) + w2*(4*(9 - sqrt(77)))^(1/3))/6 = -0.4256915364... + 0.458591887...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3))/2 are the complex roots of x^3 - 1.

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

Links

Table of n, a(n) for n=0..84.

Formula

r = ((4*(9 + sqrt(77)))^(1/3) + 4*(4*(9 + sqrt(77)))^(-1/3))/6.

r = ((4*(9 + sqrt(77)))^(1/3) + (4*(9 - sqrt(77)))^(1/3))/6.

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

Example

0.851383072866924393493940112187859385096149923938041965059002396279722...

Mathematica

RealDigits[x /. FindRoot[3*x^3 - x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)

Crossrefs

Cf. A357465, A357467.

Sequence in context: A073745 A086730 A249449 * A334496 A258988 A139721

Adjacent sequences: A357463 A357464 A357465 * A357467 A357468 A357469

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Oct 17 2022

Status

approved