A226574 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e + log(f(k-1)) for k>1.

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

Offset: 1

Author

Clark Kimberling, Jun 12 2013

Keywords

• nonn
• cons
• easy

Comments

Let g(x) be the greater of the two solutions of s + log(s) = x; then A226572 represents g(e). [See however the comments in A226571. - N. J. A. Sloane, Dec 09 2017]

Examples

			limit(f(n)) = 4.1386519474...
		

Links

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

Crossrefs

Cf. A226571, A226572, A226573.

Programs

Mathematica

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 *)

PARI

default(realprecision, 100); solve(x=4, 5, x - log(x) - exp(1)) \\ Jianing Song, Dec 24 2018

Formula

Equals -LambertW(-1, -exp(-e)). - Jianing Song, Dec 24 2018

Extensions

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