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Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x).
1

%I #24 Oct 29 2020 15:20:09

%S 4,9,8,5,6,3

%N Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x).

%C f(A199203) = 1, where f is the functional square root of exponent.

%C 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.

%H Citizendium, <a href="http://en.citizendium.org/wiki/Superfunction">Superfunction</a>.

%H Math StackExchange, <a href="https://math.stackexchange.com/questions/2135474/half-iteration-of-exponential-function">Half iteration of exponential function</a>.

%H Gottfried Helms, <a href="http://go.helms-net.de/math/tetdocs/CoefficientsForUTetration.htm">Coefficients for fractional iterates exp(x)-1</a>.

%H Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x)=F(x)</a>, arXiv:1302.1986 [math.CO], 2013.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Functional_square_root">Functional square root</a>.

%e 0.498563...

%t 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]]

%Y Cf. A001113, A184011.

%K nonn,cons,more,hard

%O 0,1

%A _Vladimir Reshetnikov_, Nov 03 2011