A256501 Decimal expansion of the positive solution to x = 4*(1-exp(-x)).

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

Offset: 1

Author

Stanislav Sykora, Apr 01 2015

Keywords

• nonn
• cons

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 = 4.

Examples

			3.9206903948728863435608913526135362205256273712079845304011750...
		

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 (q=5), A194567 (q=3), A256500 (q=2).

Programs

Mathematica

RealDigits[x/.FindRoot[x==4(1-Exp[-x]),{x,3},WorkingPrecision->120]] [[1]] (* Harvey P. Dale, May 08 2017 *)

PARI

a4=solve(x=0.1, 10, x-4*(1-exp(-x))) \\ Use real precision in excess