# Sequences with divisibility

A sequence is such that $$a_o =1, a_1= 1, a_{n+1} =a_{n}a_{n-1}+1$$ so we have to comment on divisibilty of $$a_{2007}$$ by 4.

I found out first few values in sequence as 1,1,2,3,7,22, .... which told me that only $$a_{3n}$$ is even. But can there be some other elaborative method?

The sequence $$a_n\bmod 4$$ is eventually periodic with a readily determined period ...
Hint  There are $$j with $$\,(a_j,a_{j+1})\equiv (a_k,a_{k+1})\,\pmod{\!4}\$$ since there are only finitely many such pairs $$\!\bmod {\!4}$$. The recurrence depends only on the pair of prior values so this leads to cyclic behavior.