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*The notation <math>^n a</math> means <math> a^a^\cdots^a</math> the application of exponentiation <math> n-1</math> times.
*The notation <math>^n a</math> means <math> a^a^\cdots^a</math> the application of exponentiation <math> n-1</math> times.
*For any positive real a>0 and non-negative integer <math>n\ge 0</math> we define <math>^n a</math>by:
*For any positive real a>0 and non-negative integer <math>n\ge 0</math> we define <math>^n a</math>by:
<math>^n a = \begin{cases}
<math>^n a =
\begin{cases}
1, & \mbox{if }n\mbox{=0} \\
1, & \mbox{if }n\mbox{=0} \\
a^{[(n-1)a]}, & \mbox{if }n\mbox{ >0}
a^{[(n-1)a]}, & \mbox{if }n\mbox{ >0}
\end{cases}</math>
\end{cases}</math>
f(n) =
\begin{cases}
n/2, & \mbox{if }n\mbox{ is even} \\
3n+1, & \mbox{if }n\mbox{ is odd}
\end{cases}