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A357108
Decimal expansion of the real root of 2*x^3 + x - 2.
1
8, 3, 5, 1, 2, 2, 3, 4, 8, 4, 8, 1, 3, 6, 6, 5, 1, 4, 2, 9, 1, 6, 2, 0, 0, 3, 8, 5, 9, 6, 7, 0, 2, 2, 9, 9, 1, 6, 5, 4, 1, 1, 4, 8, 7, 7, 8, 0, 4, 3, 3, 6, 0, 1, 9, 3, 6, 2, 7, 9, 7, 3, 1, 5, 3, 8, 5, 8, 9, 5, 1, 8, 1, 0, 9, 8, 0, 8
OFFSET
0,1
COMMENTS
The other (complex) roots are w1*(1/2 + (1/36)*sqrt(330))^(1/3) + (1/2 - (1/36)*sqrt(330))^(1/3) = -0.4175611742... + 1.0114702183....*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (1/6)*sqrt(6)*(-cosh((1/3)*arccosh(3*sqrt(6))) + sqrt(3)*sinh((1/3)*arccosh(3*sqrt(6)))*i), and its complex conjugate.
FORMULA
r = ((108 + 6*sqrt(330))^(1/3) - 6*(108 + 6*sqrt(330))^(-1/3))/6.
r = (6*(1/2 + (1/36)*sqrt(330))^(1/3) - (1/2 + (1/36)*sqrt(330))^(-1/3))/6.
r = (1/2 + (1/36)*sqrt(330))^(1/3) + w1*(1/2 - (1/36)*sqrt(330))^(1/3), where w1 = (-1 = sqrt(3)*i)/2 is one of the complex roots of x^3 - 1.
r= (1/3)*sqrt(6)*sinh((1/3)*arcsinh(3*sqrt(6))).
EXAMPLE
0.835122348481366514291620038596702299165411487780433601936279731538589...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 + x - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
CROSSREFS
Cf. A357107.
Sequence in context: A328498 A199440 A199293 * A110234 A334073 A196654
KEYWORD
nonn,cons,easy
AUTHOR
Wolfdieter Lang, Sep 29 2022
STATUS
approved