A273067 -id:A273067 - cp's OEIS frontend

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

Rick L. Shepherd, May 15 2016

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)

			0.565197717383639396437528013247030816098483976759553827555483810948411203...

Index entries for algebraic numbers, degree 3.

Cf. A273066, A273067, A357109.

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

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

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)

A273066 Decimal expansion of the real root of x^3 - 2x + 2, negated.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, May 14 2016

The roots of x^3 + A273065*x^2 - A273066*x + A273067 are A273065, -A273066, and A273067. The only other real, cubic, monic polynomial with nonzero constant term and equal coefficients and roots when ignoring the leading coefficient is x^3 + x^2 - x - 1 (per the Math Overflow link).

The other two roots of x^3 - 2*x - 2 are w1*(1 + (1/9)*sqrt(57))^(1/3) + w2*(1 - (1/9)*sqrt(57))^(1/3) = -0.88464617... + 0.58974280...*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 this is -(1/3)*sqrt(6)*(cosh((1/3)*arccosh((3/4)*sqrt(6))) - sqrt(3)*sinh((1/3)*arccosh((3/4)*sqrt(6)))*i) and its complex conjugate. - Wolfdieter Lang, Sep 13 2022

			1.7692923542386314152404094643350334926705530458988570042331061304026738...

A. J. Di Scala and O. Macia, Finiteness of Ulam Polynomials, arXiv:0904.0133 [math.AG], 2009.
R. Stanley, R. Israel et al., Math Overflow: Which polynomial's roots are its coefficients?, Sep 3 2015.
P. R. Stein, On Polynomial Equations with Coefficients Equal to Their Roots, The American Mathematical Monthly, Vol. 73, No. 3 (Mar., 1966), pp. 272-274.
Index entries for algebraic numbers, degree 3.

Cf. A273065, A273067, A357102.

RealDigits[Root[x^3-2x+2,1],10,120][[1]] (* Harvey P. Dale, May 25 2022 *)

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

Equals ((9-sqrt(57))^(1/3))/(3^(2/3)) + 2/((3(9-sqrt(57)))^(1/3)) (from Wolfram Alpha).
From Wolfdieter Lang, Sep 13 2022: (Start)
Equals (1 + (1/9)*sqrt(57))^(1/3) + (2/3)*(1 + (1/9)*sqrt(57))^(-1/3) [compare with the above formula which uses the negative sqrt(57)].
Equals (1 + (1/9)*sqrt(57))^(1/3) + (1 - (1/9)*sqrt(57))^(1/3).
Equals (2/3)*sqrt(6)*cosh((1/3)*arccosh((3/4)*sqrt(6))). (End)

cp's OEIS Frontend

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

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