Question 132
Let $k \gt 3$ be an integer and suppose a doubly infinite sequence of real numbers $\ldots$, $a_{-2}$, $a_{-1}$, $a_0$, $a_1$, $a_2$, $\ldots$ has the property that
$$
a_{n+k} = \frac{1}{k} \sum_{j=0}^{k-1} a_{n+j}, \quad \text{for all integers } n.
$$
If this sequence is bounded, does it follow it is constant?
$$
a_{n+k} = \frac{1}{k} \sum_{j=0}^{k-1} a_{n+j}, \quad \text{for all integers } n.
$$
If this sequence is bounded, does it follow it is constant?