Periodicity and non-periodicity

A sequence $(a_n)$ is periodic with period $p$ if $a_{n+p} = a_n$ for all $n$ after some starting point. The smallest such $p > 0$ is called the fundamental period. If no such $p$ exists, the sequence is non-periodic.

Important properties

• If a sequence is periodic with period $p$, it is also periodic with period $kp$ for any integer $k > 1$.

• A sequence can be eventually periodic: it becomes periodic after a certain point.

• Non-periodic sequences never repeat their values in a fixed cycle.