A357102 - OEIS

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A357102

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

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

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OFFSET

0,1

COMMENTS

The other two roots are (w1*(27 + 3*sqrt(105))^(1/3) + (27 - 3*sqrt(105))^(1/3))/3 = -0.3854584985... + 1.5638845105...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.

Using hyperbolic functions these roots are -(1/3)*sqrt(6)*(sinh((1/3)* arcsinh((3/4)*sqrt(6))) + sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(6)))*i), and its complex conjugate.

LINKS

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

Index entries for algebraic numbers, degree 3

FORMULA

r = (1/3)*(27 + 3*sqrt(105))^(1/3) - 2/(27 + 3*sqrt(105))^(1/3).

r = ((27 + 3*sqrt(105))^(1/3)+ w1*(27 - 3*sqrt(105))^(1/3))/3, 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)*sqrt(6)*sinh((1/3)*arcsinh((3/4)*sqrt(6))).

EXAMPLE

0.770916997059248100825146369307026967255053119363328615100598492976735103...

MATHEMATICA

RealDigits[x /. FindRoot[x^3 + 2*x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Sep 21 2022 *)

PROG