cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, May 15 2016

Keywords

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

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)

A273067 Decimal expansion of the constant term, which is also a root, of the cubic polynomial below.

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, May 15 2016

Keywords

Comments

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

Examples

			0.638896919471352622365353437840971860473585092379749349122138508505851410...

Links

Crossrefs

Cf. A273065, A273066.

Programs

Formula

A273066 - 2/A273066.

A357102 Decimal expansion of the real root of x^3 + 2*x - 2.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Sep 20 2022

Keywords

Comments

The other two roots are (w1*(27 + 3*sqrt(105))^(1/3) + (27 - 3*sqrt(105))^(1/3))/3 = -0.3854584985... + 1.5638845105...*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 -(1/3)*sqrt(6)*(sinh((1/3)* arcsinh((3/4)*sqrt(6))) + sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(6)))*i), and its complex conjugate.

Examples

			0.770916997059248100825146369307026967255053119363328615100598492976735103...

Links

Crossrefs

Cf. A273066.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^3 + 2*x - 2, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Sep 21 2022 *)
  • PARI
    solve(x=0, 1, x^3 + 2*x - 2) \\ Michel Marcus, Sep 23 2022
  • PARI
    polrootsreal(x^3 + 2*x - 2)[1] \\ Charles R Greathouse IV, Sep 30 2022

Formula

r = (1/3)*(27 + 3*sqrt(105))^(1/3) - 2/(27 + 3*sqrt(105))^(1/3).
r = ((27 + 3*sqrt(105))^(1/3)+ w1*(27 - 3*sqrt(105))^(1/3))/3, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) is one of the complex roots of x^3 - 1.
r = (2/3)*sqrt(6)*sinh((1/3)*arcsinh((3/4)*sqrt(6))).
Showing 1-3 of 3 results.

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