For the Skiponacci sequence:
$a_0=3, a_1=0, a_2=2,$ and $a_{n+1}=a_{n-1}-a_{n-2}$ for $n>2$, prove that any prime $p$ divides $a_p$.
Is there any alternate solution other than using characteristic functions and/or frobenius endomorphism? Such as an elementary solution (no advanced topics please)?