Balkan Mathematical Olympiad Shortlist

First, for $k \in \mathbb{N}$, $k \ge 2$, we have
$$a_{2k+1} + a_{2k-1} = 2a_{2k} - a_k + a_{2k-1} = 5a_{2k-1} - a_k \equiv -a_k \pmod{5}. \quad (1)$$
We prove the assertion of the problem by induction on $k$, where $n = 8k+3$. For the base case $k=1$, we compute $a_3 = 1$, $a_4 = 2$, $a_5 = 3$, $a_6 = 6$, $a_7 = 11$, $a_8 = 22$, $a_9 = 42$, $a_{10} = 84$ and $a_{11} = 165$.
By repeatedly applying (1), we have
$$\begin{align*} a_{8(k+1)+3} - a_{8k+3} &= (a_{8k+11} + a_{8k+9}) - (a_{8k+9} + a_{8k+7}) \\ &\quad +(a_{8k+7} + a_{8k+5}) - (a_{8k+5} + a_{8k+3}) \\ &\equiv -a_{4k+5} + a_{4k+4} - a_{4k+3} + a_{4k+2} \\ &\quad = -a_{4k+5} + a_{4k+3} + 2a_{4k+1} \\ &\quad = -(a_{4k+5} + a_{4k+3}) + 2(a_{4k+3} + a_{4k+1}) \\ &\equiv a_{2k+2} - 2a_{2k+1} = 0 \pmod{5}. \end{align*}$$
Therefore, if $a_{8k+3}$ is divisible by 5, so is $a_{8(k+1)+3}$. This completes the proof.