A357106 - OEIS

A357106

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

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

This equals r0 - 1/6 where r0 is the real root of y^3 - (1/12)*y - 107/108.

The complex roots of 2*x^3 + x^2 - 2 are (w1*(107 + 6*sqrt(318))^(1/3) + w2*(107 - 6*sqrt(318))^(1/3) - 1)/6 = -0.6790471647... + 0.8392067630...*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(107))) + sqrt(3)*sinh((1/3)*arccosh(107))*i)/6, and its complex conjugate.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

r = ((107 + 6*sqrt(318))^(1/3) + (107 + 6*sqrt(318))^(-1/3) - 1)/6

r = ((107 + 6*sqrt(318))^(1/3) + (107 - 6*sqrt(318))^(1/3) - 1)/6.

r = (2*cosh((1/3)*arccosh(107)) - 1)/6.

EXAMPLE

0.858094329496552706258725850958187515309026929286713666496137417447921455...

MATHEMATICA

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

RealDigits[Root[2x^3+x^2-2, 1], 10, 100][[1]] (* Harvey P. Dale, Sep 25 2023 *)