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.

A256500 Decimal expansion of the positive solution to x = 2*(1-exp(-x)).

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, Mar 31 2015

Keywords

Comments

Each of the positive solutions to x = q*(1-exp(-x)) obtained for q = 2, 3, 4, and 5, appears in several formulas pertinent to Planck's black-body radiation law. For a given q, the solution can be also written as q+W(-q/exp(q)), where W is the Lambert function. Here q = 2.
The constant appears in asymptotic formula for A007820. - Vladimir Reshetnikov, Oct 10 2016

Examples

			1.5936242600400400923230418758751602417890024248188593649995...

Links

Crossrefs

Cf. A194567 (q=3), A256501 (q=4), A256502 (q=5).
Cf. A106533, A191236, A217905, A229553.

Programs

  • Mathematica
    RealDigits[2 + LambertW[-2 Exp[-2]], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 10 2016 *)
  • PARI
    a2=solve(x=0.1,10,x-2*(1-exp(-x))) \\ Use real precision in excess

Formula

Equals 2*(1-A106533). - Miko Labalan, Dec 18 2024
Equals log(A229553). - Hugo Pfoertner, Dec 19 2024

A094090 Decimal expansion of positive solution to 5*(1-exp(u)) + u*exp(u) = 0.

Original entry on oeis.org

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

Views

Author

Jeppe Stig Nielsen, May 01 2004

Keywords

Comments

This purely mathematical constant turns up when in physics one derives Wien's displacement law from the Planck black-body radiation law (see link).
Positive solution to x = 5*(1-exp(-x)). More comments in A256500. - Stanislav Sykora, Apr 01 2015

Examples

			u=4.965114231744276...

Links

Crossrefs

Cf. A194567, A256500, A256501.

Programs

  • Mathematica
    RealDigits[5 + ProductLog[ -5/E^5], 10, 120][[1]] (* Robert G. Wilson v, May 04 2004 *)
  • PARI
    a5=solve(x=0.1, 10, x-5*(1-exp(-x))) \\ Use real precision in excess

Formula

u = 5 + W(-5*exp(-5)), where W() is Lambert's W-function.

Extensions

More terms from Robert G. Wilson v, May 04 2004

A194567 Decimal expansion of the positive solution to x = 3*(1-exp(-x)).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Aug 29 2011

Keywords

Comments

The positive solution to x=3*(1-exp(-x)) is the dimensionless coefficient corresponding to the maximum brightness in Planck's law of radiation.
It can be symbolically expressed as 3+W(-3/e^3), where W stands for Lambert (a.k.a. "ProductLog") function.

Examples

			2.821439372...

Links

Crossrefs

Cf. A094090, A256500, A256501.

Programs

  • Maple
    evalf(3+LambertW(-3/exp(3)), 130);  # Alois P. Heinz, May 08 2024
  • Mathematica
    RealDigits[ N[x /. ToRules[ Reduce[x > 0 && x == 3*(1 - E^-x), x, Reals]], 100]][[1]]
RealDigits[3 + ProductLog[-3/E^3], 10, 111][[1]] (* Robert G. Wilson v, Oct 16 2013 *)
RealDigits[x/.FindRoot[x==3(1-Exp[-x]),{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 09 2023 *)
  • PARI
    a3=solve(x=0.1,10,x-3*(1-exp(-x))) \\ Use real precision in excess
  • PARI
    3+lambertw(-3/exp(3)) \\ Charles R Greathouse IV, Sep 13 2022
Showing 1-3 of 3 results.

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