Periodic sequence

Remarks

Example: $x_n = (-1)^n$ in $\R$ satisfies $x_{n+2} = (-1)^n (-1)^2 = x_n$ for all $n$, so it is periodic, with least period $2$ since $x_1 \ne x_0$.

Non-example: $x_n = n$ is not periodic, since $x_{n+p} = n + p \ne x_n$ for every $p \ge 1$.

The sequence is eventually periodic if the condition holds for all $n \ge n_0$ for some $n_0$; for example $0, 1, 1, 1, \dots$ is eventually periodic but not periodic. Every period is a multiple of the least period. The same definition applies to two-sided sequences $(x_n)_{n \in \Z}$, with the condition required for all $n \in \Z$.