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Integer terms of a sequence

Source: Bulgarian Spring Tournament 2024 12.1

March 31, 2024
algebra

Problem Statement

Given is a sequence a1,a2,…a_1, a_2, \ldots, such that a1=1a_1=1 and an+1=9an+4an+6a_{n+1}=\frac{9a_n+4}{a_n+6} for any n∈Nn \in \mathbb{N}. Which terms of this sequence are positive integers?
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