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URL: https://oeis.org/A226571

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A226571
Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 - log(f(k-1)) for k>1.
9
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, 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, 8, 4, 0, 0, 7, 8, 5, 7, 9, 9, 5, 9, 2, 1, 5, 0, 9, 1
OFFSET
1,2
COMMENTS
Old definition was: Decimal digits of limit(f(n)), where f(1) = 2 - log(2), f(n) = f(f(n-1)).
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]
LINKS
FORMULA
Equals LambertW(exp(2)). - Vaclav Kotesovec, Jan 09 2014
EXAMPLE
2 - log 2 = 1.732378...
2 - log(2 - log 2) = 1.450504...
2 - log(2 - log(2 - log 2)) = 1.628088...
limit(f(n)) = 1.557144510523...
MATHEMATICA
f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[2, 200]][[1]] (* A226571 *)
d2 = RealDigits[g[2, 200]][[1]] (* A226572 *)
s /. NSolve[s - Log[s] == 2, 200] (* both constants *)
h[x_] := s /. NSolve[s - Log[s] == x] Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)
RealDigits[LambertW[Exp[2]], 10, 50][[1]] (* G. C. Greubel, Nov 14 2017 *)
PROG
(PARI) lambertw(exp(2)) \\ G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn,cons,easy
AUTHOR
Clark Kimberling, Jun 11 2013
EXTENSIONS
Definition edited by N. J. A. Sloane, Dec 09 2017
STATUS
approved