login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226571 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 - log(f(k-1)) for k>1. 8
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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

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

Cf. A006155, A226572, A226573, A226574.

Sequence in context: A327242 A173932 A249649 * A274030 A061382 A113272

Adjacent sequences:  A226568 A226569 A226570 * A226572 A226573 A226574

KEYWORD

nonn,cons,easy

AUTHOR

Clark Kimberling, Jun 11 2013

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)