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A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1. 10
3, 1, 4, 6, 1, 9, 3, 2, 2, 0, 6, 2, 0, 5, 8, 2, 5, 8, 5, 2, 3, 7, 0, 6, 1, 0, 2, 8, 5, 2, 1, 3, 6, 8, 2, 5, 2, 8, 8, 8, 6, 6, 2, 0, 4, 6, 1, 8, 2, 4, 8, 8, 4, 2, 6, 0, 3, 4, 6, 1, 9, 2, 9, 1, 2, 8, 6, 7, 7, 5, 1, 6, 3, 9, 8, 7, 5, 4, 8, 8, 7, 0, 7, 7, 4, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Let h(x) be the greater of the two solutions of s + log(s) = x; then this sequence represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = LambertW(exp(2)) = 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 = -LambertW(-1,-exp(-2)) = 3.14619322062... (this sequence) and s = -LambertW(-exp(-2)) = 0.158594339563... (A202348, not A226571). - Vaclav Kotesovec, Jan 09 2014]

Apart from the first digit the same as A202321. - R. J. Mathar, Jun 15 2013

LINKS

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

FORMULA

Equals -LambertW(-1, -exp(-2)) = A202321 + 2. - Vaclav Kotesovec, Jan 09 2014

EXAMPLE

2 + log 2 = 2.693147...

2 + log(2 + log 2) = 2.990710...

2 + log(2 + log(2 + log 2)) = 3.095510...

limit(f(n)) = 3.14619322062...

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 *)

PROG

(PARI) default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ Jianing Song, Dec 30 2018

CROSSREFS

Cf. A202321, A202348, A226571, A226573, A226574.

Cf. also A052526.

Sequence in context: A075419 A060922 A143790 * A251633 A135611 A306804

Adjacent sequences:  A226569 A226570 A226571 * A226573 A226574 A226575

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jun 12 2013

EXTENSIONS

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

STATUS

approved

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Last modified April 19 17:46 EDT 2021. Contains 343117 sequences. (Running on oeis4.)