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.

User: David D. Acker

David D. Acker's wiki page.

David D. Acker has authored 4 sequences.

A277077 Decimal expansion of the root of cos(sin(x)) - x = 0.

Original entry on oeis.org

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

Views

Author

David D. Acker, Sep 27 2016

Keywords

Comments

The fixed point solution for the composite function y = cos(sin(x)).
The value A131691 is equal to the arccosine of this value and this value is equal to the arcsine of A131691.

Examples

			0.76816915673679597746208623955865641813208731218273718569186715...

Crossrefs

Cf. A131691 (reversed form), A003957 (fixed point solution for cosine).

Programs

  • Mathematica
    FindRoot[-x + Cos[Sin[x]] == 0, {x, 0.5, 1}, WorkingPrecision -> 265]
  • PARI
    solve(x=0.5, 1, cos(sin(x))-x) \\ Michel Marcus, Sep 29 2016

Formula

Recursion: f(n) = cos(sin(f(n-1))) n->infinity.
Root of cos(sin(x)) - x = 0.

A277069 Decimal expansion of the real part of the root of the function x^y = y.

Original entry on oeis.org

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

Views

Author

David D. Acker, Sep 27 2016

Keywords

Comments

This value is negative.
It is not known if this number has a closed form.
This root occurs on the real component of the function x^y = y for x values which are all real.

Examples

			-99.1120147991877404695252398904774418725997764195131575672373129...

Crossrefs

Cf. A277067.

Programs

  • Mathematica
    FindRoot[Re[-ProductLog[-Log[x]]/Log[x]], {x, -100, -99}, WorkingPrecision -> 261]

Formula

Original fixed point function is of the form: x^y = y. When solved for y: y = -ProductLog(-log(x))/log(x).

A277067 Decimal expansion of value of x such that the solution y to the equation x^y = y has equal real and imaginary parts.

Original entry on oeis.org

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

Views

Author

David D. Acker, Sep 27 2016

Keywords

Comments

It is not known if this number has a closed form.

Examples

			-0.750045256460151711237852993036822415525210961075147250937205...

Crossrefs

Cf. A042972, A073229, A277069.

Programs

  • Mathematica
    FindRoot[Re[-ProductLog[-Log[x]]/Log[x]] - Im[-ProductLog[-Log[x]]/Log[x]], {x, -0.76, -0.74}, WorkingPrecision -> 261]

Formula

The solution to x^y=y is y=-ProductLog(-log(x))/log(x).

A277053 Decimal expansion of real zero x between 78 and 79 of the derivative of the function plotting the invariant points for the exponential function of the form x^y = y.

Original entry on oeis.org

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

Views

Author

David D. Acker, Sep 26 2016

Keywords

Comments

It has not yet been determined if this number has a closed form.

Examples

			78.5176688733806851928297599903919937600495951319589367155801108473527173126...

Crossrefs

Cf. A042972, A073229.

Programs

  • Mathematica
    FindRoot[Re[ProductLog[-Log[x]]^2/(x Log[x]^2 (1 + ProductLog[-Log[x]]))], {x, 78, 79},
WorkingPrecision -> 261]

Formula

The derivative x^y = y, or y = -ProductLog(-log(x))/log(x) when solved for y, is the function in which this value is a root. The derivative is (ProductLog(-log(x)))^2/(x*(log(x))^2*(1+ProductLog(-log(x)))).

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