# Evaluating the recurrence $a_{n+1}=a_n\cdot a_{n-1}-1$

Let the sequence $a_n$ be defined recursively by $a_0=3$, $a_1=1$ and $a_{n+1}=a_n\cdot a_{n-1}-1$ for $n=1,2,\cdots$. Find $a_6$, $a_7$, $a_8$, and $a_{200}$

Here is my attempt as an answer

$a_0 = 3,a_1=1,a_2=2,a_3=1,a_4=1,a_5=0,a_6=-1,a_7=-1,a_8=0$

I do not know how to find $a_{200}$

Since $a_5=0$ and $a_6=-1$, and we see this again happens at $a_8=0$ and $a_9=-1$, we know this sequence must repeat; a term only depends on the previous two values. Thus, we see the sequence is (from the fifth term on) simply $0,-1,-1$ repeated.

It is now straightforward to compute $a_{200}$.

• So if i get what you are saying $a_{200} = 3$? – Phillip Mc May 6 '17 at 23:59
• No, $a_{200}$ is not $3$. What made you think that? – vrugtehagel May 7 '17 at 0:00
• OK so i don't get it then – Phillip Mc May 7 '17 at 0:04
• As I said: what made you think that it's $3$? If you tell me what part of my post you don't fully understand, I can help appropriately. – vrugtehagel May 7 '17 at 8:03
• I would write that pattern like this: For $n\ge2$, $$a_{3n}=-1$$ $$a_{3n+1}=-1$$ $$a_{3n+2}=0$$ Now I need to decide whether $200$ is a $3n$, $3n+1$ or $3n+2$. If I know that $201$ is a $3n$ because $2+0+1=3$, then $200$ has to be... – Jaume Oliver Lafont May 13 '17 at 3:38

Perhaps a different point of view would help. Suppose we define a mapping $f(x,y) := (y,xy-1)$ and $P_{n+1} := f(P_n)$ where $P_0 = (a_0,a_1)$. It is easy to see, by induction, that $P_n = (a_n,a_{n+1})$ for all nonnegative $n$. Such a simple mapping $f$ has complicated behavior, but it may have limit cycles. That is, if we can find $i<j$ such that $P_i = P_j$, then the sequence $P$ will repeat indefinitely from $i$ on. That is, going forward, from $i$ on, the only values of $P_n$ will be $P_i,P_{i+1},...,P_{j-1}$ cyclicly repeated. Thus, to find $P_n$, from $i$ on, you only need to know where in the cycle $n$ corresponds. In your problem, I suggest looking at $P_0, P_3, P_6,P_9,\dots$ and I hope this explanation is more understandable.