$$a(n)=a(\lceil \mathop{\rm abs}(a(n-1)) \rceil\bmod n)) + a(\lceil \mathop{\rm abs}(a(n-2)) \rceil\bmod n))$$ For starting values, $a(0)=a(1)=1$, the sequence has a cycle starting with $n=441329$ having a period of $63584$ (source: https://oeis.org/A330615) For starting value $a(0)=i$ and $a(1)=1$, the cycle starts at $n=35694$ and have a period of $3605$. My conjecture is that for any complex starting values, this sequence eventually cycles. Can anybody prove or disprove this conjecture?
Newer conjecture: The sequence either cycles or eventually forms a arithmetic progression
For starting values $a(0) = 1$ and $a(1) = 2$, the sequence simply satisfies $a(n) = n+1$ and does not eventually cycle; hence the conjecture is false.