A357463 Decimal expansion of the real root of 2*x^3 + 2*x - 1.

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

Offset: 0

Author

Wolfdieter Lang, Sep 29 2022

Keywords

• nonn
• cons
• easy

Comments

The other (complex) roots are w1*((1 + (1/9)*sqrt(129))/4)^(1/3) + ((1 - (1/9)*sqrt(129))/4)^(1/3) = -0.2119268995... + 1.0652413023...*i, and its complex conjugate, where w1 = (-1 + sqrt(3))/2 = exp((2/3)*Pi*i).

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

Examples

			0.423853799069783271378040062625515233676388197185177540823008396819954728...
		

Crossrefs

Cf. A316711 (Comment).

Programs

Mathematica

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

Formula

r = ((1 +(1/9)*sqrt(129))/4)^(1/3) - (1/3)*((1 + (1/9)*sqrt(129))/4)^(-1/3).

r = ((1 + (1/9)*sqrt(129))/4)^(1/3) + w1*((1 - (1/9)*sqrt(129))/4)^(1/3), where w1 = (-1 + sqrt(3))/2, one of the complex roots of x^3 - 1.

r = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).