a_{n+1} \leq a_{n+2} \Leftrightarrow a_n^2 + \displaystyle\frac{1}{6}a_{n-1} + \displaystyle\frac{1}{9} \leq a_{n+1}^2 + \displaystyle\frac{1}{6}a_n + \displaystyle\frac{1}{9} \Leftrightarrow a_{n}^2 + \displaystyle\frac{1}{6}a_{n-1} \leq a_{n+1}^2 + \displaystyle\frac{1}{6}a_{n}