A357469 Decimal expansion of the real root of x^3 - x^2 + x - 2.

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

Offset: 1

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 - 47/27, after 1/3.

The other (complex) roots of x^3 - x^2 + x - 2 are (w1*(4*(47 + 3*sqrt(249)))^(1/3) + (4*(47 - 3*sqrt(249)))^(1/3) + 2)/6 = -0.1766049820... + 1.2028208192...*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 (-sqrt(2)*(sinh((1/3)*arcsinh((47/8)*sqrt(2))) - sqrt(3)*cosh((1/3)*arcsinh((47/8)*sqrt(2)))*i) + 1)/3, and its complex conjugate.

Examples

			1.3532099641993244294831013325773884572707056138568468268066930426515189723220920859165...
		

Links

• Index entries for algebraic numbers, degree 3

Crossrefs

Cf. A197032, A137421, A357468.

Programs

Maple

Digits := 140 ;
r := (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3 ;
evalf(%) ; # R. J. Mathar, Nov 08 2022

Mathematica

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

Formula

r = ((4*(47 + 3*sqrt(249)))^(1/3) - 8*(4*(47 + 3*sqrt(249)))^(-1/3) + 2)/6.

r = ((4*(47 + 3*sqrt(249)))^(1/3) + w1*(4*(47 - 3*sqrt(249)))^(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 = (2*sqrt(2)*sinh((1/3)*arcsinh((47/8)*sqrt(2))) + 1)/3.

Equals A197032 minus one.