cp's OEIS Frontend

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

%I A194567 #29 Nov 21 2024 15:35:09
%S A194567 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,
%T A194567 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,
%U A194567 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
%N A194567 Decimal expansion of the positive solution to x = 3*(1-exp(-x)).
%C A194567 The positive solution to x=3*(1-exp(-x)) is the dimensionless coefficient corresponding to the maximum brightness in Planck's law of radiation.
%C A194567 It can be symbolically expressed as 3+W(-3/e^3), where W stands for Lambert (a.k.a. "ProductLog") function.
%H A194567 Stanislav Sykora, <a href="/A194567/b194567.txt">Table of n, a(n) for n = 1..2000</a>
%H A194567 SpectralCalc, <a href="http://www.spectralcalc.com/blackbody/blackbody.html">Calculation of Blackbody Radiance</a>, Appendix C.
%H A194567 Wikipedia, <a href="http://en.wikipedia.org/wiki/Planck%27s_law">Planck's law</a>
%H A194567 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%e A194567 2.821439372...
%p A194567 evalf(3+LambertW(-3/exp(3)), 130);  # _Alois P. Heinz_, May 08 2024
%t A194567 RealDigits[ N[x /. ToRules[ Reduce[x > 0 && x == 3*(1 - E^-x), x, Reals]], 100]][[1]]
%t A194567 RealDigits[3 + ProductLog[-3/E^3], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 16 2013 *)
%t A194567 RealDigits[x/.FindRoot[x==3(1-Exp[-x]),{x,2},WorkingPrecision->120]][[1]] (* _Harvey P. Dale_, Aug 09 2023 *)
%o A194567 (PARI) a3=solve(x=0.1,10,x-3*(1-exp(-x))) \\ Use real precision in excess
%o A194567 (PARI) 3+lambertw(-3/exp(3)) \\ _Charles R Greathouse IV_, Sep 13 2022
%Y A194567 Cf. A094090, A256500, A256501.
%K A194567 nonn,cons
%O A194567 1,1
%A A194567 _Jean-François Alcover_, Aug 29 2011