A357104 Decimal expansion of the real root of x^3 + 3*x - 1.

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

Offset: 0

Author

Wolfdieter Lang, Sep 21 2022

Keywords

• nonn
• cons
• easy

Comments

The other two roots are w1*phi^(1/3) - w2*(-1 + phi)^(1/3) = -0.16109267... + 1.75438095...*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; phi = A001622.

The hyperbolic function version is -sinh((1/3)*arcsinh(1/2)) + sqrt(3)*cosh((1/3)*arcsinh(1/2))*i, and its complex conjugate.

Examples

			0.32218535462608559291147071070403198493164438289958400917884391190429676...
		

Crossrefs

Cf. A001622, A130880, A332437, A332438 - 2.

Programs

Maple

h := ((1 + sqrt(5))/2)^(1/3): evalf(h - 1/h, 90); # Peter Luschny, Sep 24 2022

Mathematica

RealDigits[Subtract @@ Surd[GoldenRatio, {3, -3}], 10, 100][[1]] (* Amiram Eldar, Sep 21 2022 *)
RealDigits[Root[x^3+3x-1,1],10,120][[1]] (* Harvey P. Dale, Oct 09 2023 *)

PARI

2*sinh((1/3)*asinh(1/2)) \\ Michel Marcus, Sep 23 2022

Formula

r = phi^(1/3) - phi^(-1/3), with phi = A001622.

r = phi^(1/3) - (-1 + phi)^(1/3).

r = 2*sinh((1/3)*arcsinh(1/2)).