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 A199203 Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x). 1
 4, 9, 8, 5, 6, 3 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS f(A199203) = 1, where f is the functional square root of exponent. The listed digits are a conjecture obtained as a common result of several different numeric approximation algorithms, they haven't been rigorously proved to be correct. LINKS Table of n, a(n) for n=0..5. Citizendium, Superfunction. Math StackExchange, Half iteration of exponential function. Gottfried Helms, Coefficients for fractional iterates exp(x)-1. Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013. Wikipedia, Functional square root. EXAMPLE 0.498563... MATHEMATICA n = 23; p[s_, 1] := c[s]; p[0, n_] := c[0]^n; p[s_, n_] := p[s, n] = Expand[Sum[c[k] p[s - k, n - 1], {k, 0, s}]]; d[n_, 0] := Sum[c[k] c[0]^k, {k, 0, n}]; d[n_, i_] := Sum[c[k] p[i, k], {k, 1, n}]; a = c[0] /. FindRoot[Table[d[n, k] k! == 1, {k, 0, n}], Table[{c[k], 2^-k}, {k, 0, n}], WorkingPrecision -> 30]; First[RealDigits[a, 10, 6]] CROSSREFS Cf. A001113, A184011. Sequence in context: A198548 A134902 A110992 * A370565 A370567 A371500 Adjacent sequences: A199200 A199201 A199202 * A199204 A199205 A199206 KEYWORD nonn,cons,more,hard AUTHOR Vladimir Reshetnikov, Nov 03 2011 STATUS approved

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Last modified July 25 03:25 EDT 2024. Contains 374586 sequences. (Running on oeis4.)