3.0 Algebra Number Theory

Let \( a_n \) be a sequence defined by \( a_1=1, a_2=2 \), and \( a_{n+2} = 2a_{n+1} + a_n \) for \( n \ge 1 \).
Prove by induction that \( a_n \) is even if and only if \( n \) is a multiple of 3.

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