Problem

Source: III International Festival of Young Mathematicians Sozopol 2012, Theme for 10-12 grade

Tags: number theory, Sequences, Sum of powers



We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.