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All good sequences are degenerate

Source: EGMO 2022/3

April 10, 2022
EGMOEGMO2022

Problem Statement

An infinite sequence of positive integers a1,a2,…a_1, a_2, \dots is called goodgood if (1) a1a_1 is a perfect square, and (2) for any integer n≥2n \ge 2, ana_n is the smallest positive integer such that na1+(n−1)a2+⋯+2an−1+anna_1 + (n-1)a_2 + \dots + 2a_{n-1} + a_n is a perfect square. Prove that for any good sequence a1,a2,…a_1, a_2, \dots, there exists a positive integer kk such that an=aka_n=a_k for all integers n≥kn \ge k. (reposting because the other thread didn't get moved)
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