Let $(a_n)$ and $(b_n)$ be sequences of real numbers such that $(1+2i{)}^n=a_n+b_{n}i$ for all integers $n\ge 0$, where $i=\sqrt{-1}$. What is the value of ${\sum }_{n=0}^{\infty }\frac{a_n^2-b_n^2}{6^n}$?