A226573 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e - log(f(k-1)) for k>1.

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

Offset: 1

Author

Clark Kimberling, Jun 12 2013

Keywords

• nonn
• cons
• easy

Comments

Old definition was: Decimal digits of limit(f(n)), where f(1) = e-log(e), f(n) = f(f(n-1)).

Let f(x) be lesser of the two solutions of s - log(s) = x; then A226571 represents f(e). [See however the comments in A226571. - N. J. A. Sloane, Dec 09 2017]

Examples

			limit(f(n)) = 2.0167797639...
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A226571, A226572, A226574.

Programs

Mathematica

Program 1:
f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[E, 200]][[1]]  (* A226573 *)
d2 = RealDigits[g[E, 200]][[1]]  (* A226574 *)
s /. NSolve[s - Log[s] == 2, 200] (* both constants *)
***
Program 2:
N[ProductLog[E^E], 100] (* Clark Kimberling, Feb 15 2018 *)

PARI

default(realprecision, 100); lambertw(exp(exp(1))) \\ G. C. Greubel, Sep 09 2018

Formula

Equals LambertW(e^e). - Clark Kimberling, Feb 15 2018

Extensions

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