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Show every term in sequence is an integer

Source: Spanish MO 2012 Q5

June 7, 2012
algebra unsolvedalgebra

Problem Statement

A sequence (an)n≥1(a_n)_{n\ge 1} of integers is defined by the recurrence a1=1, a2=5, an=an−12+4an−2 for n≥2.a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2. Prove that all terms of the sequence are integers and find an explicit formula for ana_n.
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