A199203 Decimal expansion of f(0) where f is the functional square root (half-iterate) of exponent, f(f(x))=exp(x).

4, 9, 8, 5, 6, 3

Offset: 0

Author

Vladimir Reshetnikov, Nov 03 2011

Keywords

• nonn
• cons
• more
• hard

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.

Examples

			0.498563...
		

Links

• 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.

Crossrefs

Cf. A001113, A184011.

Programs

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]]