GGBNA 2012 #8

This is a 2-dimensional representation of the iteration rule $a_{n+2} = \frac{a_{n+1}+k}{a_n}$. Now, instead of always adding 1 to $a_{n+1}$, we can add something other than 1! The points plotted are of the form $\left(a_n,a_{n+1} \right)$. The coordinates of point A are $\left(a_1,a_2 \right)$, so moving point A changes the value of the first two terms of the sequence. Click on point A and use the arrow keys on your keyboard to move point A.

Just observe the patterns before you decide to trace. Do there appear to be any other "nice" orbits (like our orbit of 5)? What happens when $k$ is negative? What happens when $k$ is zero? Do there appear to be any special values of $k$?