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

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

Author

Jean-François Alcover, Aug 29 2011

Keywords

• nonn
• cons

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

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
• Wikipedia, Planck's law
• Index entries for transcendental numbers

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