A357467 Decimal expansion of the real root of 3*x^3 + x - 1.

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

Offset

0,1

Comments

The other roots are w1*(4*(9 + sqrt(85)))^(1/3) + ((4*(9 - sqrt(85)))^(1/3)))/6 = -0.2682825823... + 0.741120749...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex roots of x^3 - 1.

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

Links

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

Formula

r = ((4*(9 + sqrt(85)))^(1/3) - 4*(4*(9 + sqrt(85)))^(-1/3))/6.

r = ((4*(9 + sqrt(85)))^(1/3) + w1*((4*(9 - sqrt(85)))^(1/3)))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.

r = (2/3)*sinh((1/3)*arcsinh(9/2)).

Example

0.5365651646722229187574245122387738338212422637521880663142371514206...

Mathematica

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

Crossrefs

Cf. A357464, A357466.

Sequence in context: A075264 A221710 A011504 * A333009 A363229 A237656

Adjacent sequences: A357464 A357465 A357466 * A357468 A357469 A357470

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Oct 17 2022

Status

approved