We define the function $f : \mathbb{N} \longrightarrow \mathbb{N} $ (where $\mathbb{N}$ excludes zero) such that

$$ f(a) = \left\{ \begin{array}{rl} 3a+1 & \text{ if } a \text{ is odd } \\ a/2 & \text{ if } a \text{ is even.} \end{array} \right. $$

Then let the sequence $c_n$ be defined as $c_i = f(c_{i-1})$ with $c_0$ an arbitrary natural seed value.

It is conjectured that the sequence $c_0, c_1, c_2, \ldots$ will always end in $1,4,2$ repeating infinitely. This has been verified by computer up to very large values of $c_0$ but is unproven in general. It is also not known whether this problem is decideable. This is generally called the Collatz problem.

The sequence $c_n$ is sometimes called the ``hailstone sequence''. This is because it behaves analogously to a hailstone in a cloud which falls by gravity and is tossed up again repeatedly. The sequence similarly ends in an eternal oscillation.