A273065 Decimal expansion of the negative reciprocal of the real root of x^3 - 2x + 2.

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

Offset: 0

Author

Rick L. Shepherd, May 15 2016

Keywords

• nonn
• cons

Comments

The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. See A273066, the main entry.

From Wolfdieter Lang, Sep 15 2022: (Start)

This equals the real root of 2*x^3 + 2*x^2 - 1, that is the real root of y^3 - (1/3)*y - 23/54, after subtracting 1/3.

The other two roots of 2*x^3 + 2*x^2 - 1 are (w1*(23/4 + (3/4)*sqrt(57))^(1/3) + w2*(23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3 = -0.7825988586... + 0.5217137179...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.

Using hyperbolic functions these roots are -((1 + cosh((1/3)*arccosh(23/4)) + sqrt(3)*sinh((1/3)*arccosh(23/4))*i)/3) and its complex conjugate. (End)

Examples

			0.565197717383639396437528013247030816098483976759553827555483810948411203...
		

Links

• Index entries for algebraic numbers, degree 3.

Crossrefs

Cf. A273066, A273067, A357109.

Programs

Mathematica

First[RealDigits[1/x/.N[First[Solve[x^3-2x+2==0,x]],105]]] (* Stefano Spezia, Sep 15 2022 *)

PARI

default(realprecision, 200);
-1/solve(x = -1.8, -1.7, x^3 - 2*x + 2)

Formula

Equals 1/A273066.

From Wolfdieter Lang, Sep 15 2022: (Start)

Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 + (3/4)*sqrt(57))^(-1/3) - 1)/3.

Equals ((23/4 + (3/4)*sqrt(57))^(1/3) + (23/4 - (3/4)*sqrt(57))^(1/3) - 1)/3.

Equals (2*cosh((1/3)*arccosh(23/4)) - 1)/3. (End)