Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.
Source: MOP 2006 Homework - Black Group
Tags: algebra unsolved, algebra
Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.