AT-OEMO-REG 2014 Problem 3

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$ . (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.