CN-TST 2012 Problem 9

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$ , let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$ , where $1\le i<j\le n-1$ . Given that there are only a finite number of even numbers in $\{a_n\}$ , prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$ , we have $$a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.$$