\begin{definition}[Definition:Collatz Sequence]
Let $f: \N \to \N$ be the mapping defined on the natural numbers as follows:
:$\forall n \in \N: \map f n = \begin{cases}
n / 2 & : n \text { even} \\
3 n + 1 & : n \text { odd} \end{cases}$
For any given value of $n$, let the sequence $\sequence {S_k}$ be defined:
:$\forall k \in \N: S_k = \begin {cases}
n & : k = 0 \\
\map f {S_{k - 1} } & : k > 0 \end {cases}$
Then $\sequence {S_k}$ is known as a '''Collatz sequence'''.
\end{definition}