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

%I A256500 #31 Dec 19 2024 02:59:26
%S A256500 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,
%T A256500 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,
%U A256500 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
%N A256500 Decimal expansion of the positive solution to x = 2*(1-exp(-x)).
%C A256500 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.
%C A256500 The constant appears in asymptotic formula for A007820. - _Vladimir Reshetnikov_, Oct 10 2016
%H A256500 Stanislav Sykora, <a href="/A256500/b256500.txt">Table of n, a(n) for n = 1..2000</a>
%H A256500 V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 249.
%H A256500 SpectralCalc, <a href="http://www.spectralcalc.com/blackbody/blackbody.html">Calculation of Blackbody Radiance</a>, Appendix C.
%H A256500 Wikipedia, <a href="http://en.wikipedia.org/wiki/Planck%27s_law">Planck's law</a>
%H A256500 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F A256500 Equals 2*(1-A106533). - _Miko Labalan_, Dec 18 2024
%F A256500 Equals log(A229553). - _Hugo Pfoertner_, Dec 19 2024
%e A256500 1.5936242600400400923230418758751602417890024248188593649995...
%t A256500 RealDigits[2 + LambertW[-2 Exp[-2]], 10, 100][[1]] (* _Vladimir Reshetnikov_, Oct 10 2016 *)
%o A256500 (PARI) a2=solve(x=0.1,10,x-2*(1-exp(-x))) \\ Use real precision in excess
%Y A256500 Cf. A194567 (q=3), A256501 (q=4), A256502 (q=5).
%Y A256500 Cf. A106533, A191236, A217905, A229553.
%K A256500 nonn,cons
%O A256500 1,2
%A A256500 _Stanislav Sykora_, Mar 31 2015