Question 59
Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences defined recursively by $a_{0}=2 ; b_{0}=2$, and $a_{n+1}=a_{n} \sqrt{1+a_{n}^{2}+b_{n}^{2}}-b_{n}$; $b_{n+1}=b_{n} \sqrt{1+a_{n}^{2}+b_{n}^{2}}+a_{n}$. Find the ternary (base 3) representation of $a_{4}$ and $b_{4}$.