Prove or refute: $a[n]=\lfloor(n+1)\pi\rfloor-\lfloor n\pi\rfloor$ is periodic.
This sequence looks periodic, starting with ${3, 3, 3, 3, 3, 3, 4, \,3, 3, 3, 3, 3, 3, 4,\, 3, 3, 3, 3, 3, 3, 4,\dots}$. After a while, there is a sequence of seven threes: $\dots, 3, 3, 3, 3, 3, 3, 3, 4,\dots$ which disproves the period being $7$.
However, I believe it's not periodic, not even for larger periods. How could I prove it?