A358184 Decimal expansion of the real root of 2*x^3 - x^2 + x - 1.

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

Offset: 0

Author

Wolfdieter Lang, Nov 07 2022

Keywords

• nonn
• cons
• easy

Comments

This equals r0 - 1/6 where r0 is the real root of y^3 - (7/12)*y - 11.

The other (complex) roots of 2*x^3 - x^2 + x - 1 are (1 + w1*(46 + 3*sqrt(249))^(1/3) + (46 - 3*sqrt(249))^(1/3))/6 = -0.1194918107... + 0.8138345589...*i, and its 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((46/25)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((46/25)*sqrt(5)))*i))/6, and its complex conjugate.

Examples

			0.73898362150450623732346254067108755072377477637909672211795496930230203...
		

Crossrefs

Cf. A358182, A358183.

Programs

Mathematica

RealDigits[x /. FindRoot[2*x^3 - x^2 + x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Nov 08 2022 *)
RealDigits[Root[-1+x-x^2+2 x^3,1],10,120][[1]] (* Harvey P. Dale, Sep 09 2023 *)

Formula

r = (1 + (46 + 3*sqrt(249))^(1/3) - 5*(46+3*sqrt(249))^(-1/3))/6.

r = (1+ (46 + 3*sqrt(249))^(1/3) + w1*(46 - 3*sqrt(249))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3).

r = (1 + 2*sqrt(5)*sinh((1/3)*arcsinh((46/25)*sqrt(5))))/6.

r = (1/6) + (46/45)*Hyper2F1([1/3, 2/3],[3/2], -(46^2/5^3)). - Gerry Martens, Nov 08 2022