A357105 Decimal expansion of the real root of 2*x^3 - x^2 - 2.

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

Offset

1,3

Comments

This equals r0 + 1/6 where r0 is the real root of y^3 - (1/12)*y - 109/108.

The complex roots of 2*x^3 - x^2 - 2 are (w1*(109 + 6*sqrt(330))^(1/3) + w2*(109 - 6*sqrt(330))^(1/3) + 1)/6 = -0.3487146684... + 0.8447013842...*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 (-cosh((1/3)*arccosh(109)) + sqrt(3)*sinh((1/3)*arccosh(109))*i)/6, and its complex conjugate.

Links

Table of n, a(n) for n=1..78.

Formula

r = ((109 + 6*sqrt(330))^(1/3) + (109 + 6*sqrt(330))^(-1/3) + 1)/6.

r = ((109 + 6*sqrt(330))^(1/3) + (109 - 6*sqrt(330))^(1/3) + 1)/6.

r = (2*cosh((1/3)*arccosh(109)) + 1)/6.

Example

1.197429336933032971559300287794721737140756086323958649381751358853315707...

Mathematica

RealDigits[x /. FindRoot[2*x^3 - x^2 - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)

Crossrefs

Cf. A357106.

Sequence in context: A092425 A019647 A318437 * A011115 A019886 A335564

Adjacent sequences: A357102 A357103 A357104 * A357106 A357107 A357108

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Sep 29 2022

Status

approved