so just define two sequences $b, c$ the even and odd terms so that $b_{n+3} = b_{n+2} + b_{n+1} - b_n$, similar one for $c$. Then characteristic equation is $r^3 - r^2 - r + 1 = 0 \Rightarrow (r-1)(r^2-1) = 0 \Rightarrow r = -1, 1, 1$ so $b_n = \alpha + n\beta + \gamma (-1)^n$; if $\beta \ne 0$, then for $n$ arbitrarily large, the modulus will be unbounded, which cannot happen, so $\beta = 0$ and $b_n = \alpha + \gamma (-1)^n$ which has period 2; a similar analysis takes place for the $c_n$ so $a_n$ has period $4$.