A357101 - OEIS

A357101

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

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

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OFFSET

1,1

COMMENTS

This equals r0 + 2/3 where r0 is the real root of y^3 - (4/3)*y - 70/27.

The other two roots are (w1*(35 + 3*sqrt(129))^(1/3) + w2*(35 - 3*sqrt(129))^(1/3 + 2)/3 = -0.1796520429... + 0.9030131458...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are (2/3)*(1 - cosh((1/3)*arccosh(35/8)) + sqrt(3)*sinh((1/3)*arccosh(35/8))*i), and its complex conjugate.

LINKS

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

Index entries for algebraic numbers, degree 3

FORMULA

r = ((35 + 3*sqrt(129))^(1/3) + 4*(35 + 3*sqrt(129))^(-1/3) + 2)/3.

r = ((35 + 3*sqrt(129))^(1/3) + (35 - 3*sqrt(129))^(1/3) + 2)/3.

r = (2/3)*(2*cosh((1/3)*arccosh(35/8)) + 1).

EXAMPLE

2.359304085971776420730660392800532555346481278067672283797141251583875588...

MAPLE

h := ((35 + 3*sqrt(129))/8)^(1/3): evalf((1 + h + 1/h)*2/3, 82); # Peter Luschny, Sep 25 2022