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A226572
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Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.
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10
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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
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OFFSET
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1,1
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COMMENTS
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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]
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LINKS
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FORMULA
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EXAMPLE
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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...
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MATHEMATICA
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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 *)
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PROG
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(PARI) default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ Jianing Song, Dec 30 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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