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

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

Author

Stanislav Sykora, Mar 31 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 = 2.

The constant appears in asymptotic formula for A007820. - Vladimir Reshetnikov, Oct 10 2016

Examples

			1.5936242600400400923230418758751602417890024248188593649995...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 249.
• SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
• Wikipedia, Planck's law
• Index entries for transcendental numbers

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