Does each term of this sequence equal to zero?

Question

We define a sequence by recursion: $$a_{2n}=a_{n}+a_{3n-1}$$ $$a_{3n}=a_{2n-1}+a_{4n+3}$$ $$a_{4n}=a_{3n-2}+a_{2n+1}-a_{n+7}$$

Question: Does each term of this sequence equal to zero?

I have proved $a_1=a_2=a_3=...=a_{17}=0$.