A357471 Decimal expansion of the real root of x^3 - x^2 + 2*x - 1.

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

Offset: 0

Author

Wolfdieter Lang, Oct 25 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 + 1/3 where r0 is the real root of y^3 + (5/3)*y - 11/27.

The other roots of x^3 - x^2 + 2*x - 1 are (1 + w1*((11 + 3*sqrt(69))/2)^(1/3) + ((11 - 3*sqrt(69))/2)^(1/3))/3 = 0.2150798545... + 1.3071412786...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex conjugate roots of x^3 - 1.

Using hyperbolic functions these roots are (1 - sqrt(5)*(sinh((1/3)*arcsinh((11/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((11/50)*sqrt(5)))*i))/3, and its complex conjugate.

Examples

			0.569840290998053265911399958119568648839797439128940220544731079656747...
		

Crossrefs

Cf. A160389, A255524, A255249, A357470, A357472.

Programs

Mathematica

RealDigits[x /. FindRoot[x^3 - x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)

Formula

r = (2 + (4*(11 + 3*sqrt(69)))^(1/3) - 20*(4*(11 + 3*sqrt(69)))^(-1/3))/6.

r = (2 + (4*(11 + 3*sqrt(69)))^(1/3) + w1*(4*(11 - 3*sqrt(69)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex conjugate roots of x^3 - 1.

r = (1 + 2*sqrt(5)*sinh((1/3)*arcsinh((11/50)*sqrt(5))))/3.

r = (1/3) + (11/45)*Hyper2F1([1/3, 2/3],[3/2], -(11^2)/(2^2*5^3)). - Gerry Martens, Nov 04 2022