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JBMO Shortlist 2020 A2

Source: JBMO Shortlist 2020

July 4, 2021
JuniorBalkanshortlist2020algebraSequence

Problem Statement

Consider the sequence a1,a2,a3,...a_1, a_2, a_3, ... defined by a1=9a_1 = 9 and
an+1=(n+5)an+22n+3a_{n + 1} = \frac{(n + 5)a_n + 22}{n + 3}
for n≥1n \ge 1.
Find all natural numbers nn for which ana_n is a perfect square of an integer.
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