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periodic sequence

Source: 2012 China TST Test 2 p3

March 19, 2012
pigeonhole principlecombinatorics proposedcombinatorics

Problem Statement

Let a1<a2a_1<a_2 be two given integers. For any integer n≥3n\ge 3, let ana_n be the smallest integer which is larger than an−1a_{n-1} and can be uniquely represented as ai+aja_i+a_j, where 1≤i<j≤n−11\le i<j\le n-1. Given that there are only a finite number of even numbers in {an}\{a_n\}, prove that the sequence {an+1−an}\{a_{n+1}-a_{n}\} is eventually periodic, i.e. that there exist positive integers T,NT,N such that for all integers n>Nn>N, we have aT+n+1−aT+n=an+1−an.a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.
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