A226571 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 - log(f(k-1)) for k>1.

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

Author

Clark Kimberling, Jun 11 2013

Keywords

• nonn
• cons
• easy

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]

Examples

			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...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006155, A226572, A226573, A226574.

Programs

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 *)

PARI

lambertw(exp(2)) \\ G. C. Greubel, Nov 14 2017

Formula

Equals LambertW(exp(2)). - Vaclav Kotesovec, Jan 09 2014

Extensions

Definition edited by N. J. A. Sloane, Dec 09 2017