Possible Duplicate:
Recurrence relation, Fibonacci numbers
$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$.
$(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ are coprime for $n \in \mathbb N$
$(ii)$ Assume that the set $\{a_n , a_{n+1} , a_{n+2}\}$ is pairwise coprime for $n \in \mathbb N$. Prove that all $a_n$ are integers by induction.
$(b)$ Consider the recurrence relation $a_{n+2}a_n = a^2_{n+1} + 1$ with $a_1 = 1, a_2 = 2$ and compare this sequence to the Fibonacci numbers. What do you find? Formulate it as a mathematical statement and prove it.