A226572 - OEIS

%I #31 Dec 31 2018 14:09:56

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

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

%U 7,5,1,6,3,9,8,7,5,4,8,8,7,0,7,7,4,3

%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 Let h(x) be the greater of the two solutions of s + log(s) = x; then this sequence represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = LambertW(exp(2)) = 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 = -LambertW(-1,-exp(-2)) = 3.14619322062... (this sequence) and s = -LambertW(-exp(-2)) = 0.158594339563... (A202348, not A226571). - _Vaclav Kotesovec_, Jan 09 2014]

%C Apart from the first digit the same as A202321. - _R. J. Mathar_, Jun 15 2013

%H Clark Kimberling, <a href="/A226572/b226572.txt">Table of n, a(n) for n = 1..1000</a>

%F Equals -LambertW(-1, -exp(-2)) = A202321 + 2. - _Vaclav Kotesovec_, Jan 09 2014

%e 2 + log 2 = 2.693147...

%e 2 + log(2 + log 2) = 2.990710...

%e 2 + log(2 + log(2 + log 2)) = 3.095510...

%e limit(f(n)) = 3.14619322062...

%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]

%t 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 *)

%o (PARI) default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ _Jianing Song_, Dec 30 2018

%Y Cf. A202321, A202348, A226571, A226573, A226574.

%Y Cf. also A052526.

%K nonn,cons

%O 1,1

%A _Clark Kimberling_, Jun 12 2013

%E Definition revised by _N. J. A. Sloane_, Dec 09 2017