A357468 Decimal expansion of the real root of x^3 + x^2 + x - 2.

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

Offset: 0

Author

Wolfdieter Lang, Oct 17 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 - 1/3 where r0 is the real root of y^3 + (2/3)*y - 61/27.

The other roots of x^3 + x^2 + x - 2 are (w1*(4*(61 + 3*sqrt(417)))^(1/3) + (4*(61 - 3*sqrt(417)))^(1/3) - 2)/6 = -0.9052678568... + 1.2837421720...*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)*(1 + sqrt(2)*(sinh((1/3)*arcsinh((61/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((61/8)*sqrt(2)))*i)), and its complex conjugate.

Examples

			0.8105357137661367740212514143256682141072614900005302474430976745094594...
		

Crossrefs

Cf. A137421.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 + x^2 + x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 18 2022 *)

Formula

r = ((4*(61 + 3*sqrt(417)))^(1/3) - 8*(4*(61 + 3*sqrt(417)))^(-1/3) - 2)/6.

r = ((4*(61 + 3*sqrt(417)))^(1/3) + w1*(4*(61 - 3*sqrt(417)))^(1/3) - 2)/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.

r = (-1 + 2*sqrt(2)*sinh((1/3)*arcsinh((61/8)*sqrt(2))))/3.