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%I #25 Jun 06 2019 08:58:46
%S 1,5,5,7,1,4,5,5,9,8,9,9,7,6,1,1,4,1,6,8,5,8,6,7,2,0,0,0,0,0,0,6,6,3,
%T 1,8,0,2,8,3,7,3,7,8,7,0,6,2,6,5,2,0,3,1,5,2,8,2,2,6,6,9,2,3,0,1,7,9,
%U 8,4,0,0,7,8,5,7,9,9,5,9,2,1,5,0,9,1
%N Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 - log(f(k-1)) for k>1.
%C Old definition was: Decimal digits of limit(f(n)), where f(1) = 2 - log(2), f(n) = f(f(n-1)).
%C Let h(x) be lesser of the two solutions of s - log(s) = x; then A226571 represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = 1.5571455989976... is the unique root of the equation s + log(s) = 2. Equation s - log(s) = 2 does have two roots, but they are s = 3.14619322062... (=A226572) and s = 0.158594339563... (not A226571). - _Vaclav Kotesovec_, Jan 09 2014]
%H Clark Kimberling, <a href="/A226571/b226571.txt">Table of n, a(n) for n = 1..1000</a>
%F Equals LambertW(exp(2)). - _Vaclav Kotesovec_, Jan 09 2014
%e 2 - log 2 = 1.732378...
%e 2 - log(2 - log 2) = 1.450504...
%e 2 - log(2 - log(2 - log 2)) = 1.628088...
%e limit(f(n)) = 1.557144510523...
%t f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
%t g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
%t d1 = RealDigits[f[2, 200]][[1]] (* A226571 *)
%t d2 = RealDigits[g[2, 200]][[1]] (* A226572 *)
%t s /. NSolve[s - Log[s] == 2, 200] (* both constants *)
%t h[x_] := s /. NSolve[s - Log[s] == x] Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
%t Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)
%t RealDigits[LambertW[Exp[2]], 10, 50][[1]] (* _G. C. Greubel_, Nov 14 2017 *)
%o (PARI) lambertw(exp(2)) \\ _G. C. Greubel_, Nov 14 2017
%Y Cf. A006155, A226572, A226573, A226574.
%K nonn,cons,easy
%O 1,2
%A _Clark Kimberling_, Jun 11 2013
%E Definition edited by _N. J. A. Sloane_, Dec 09 2017
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