IMO Shortlist 1988 problem 1

An integer sequence is defined by

$a_n = 2 \cdot a_{n-1} + a_{n-2}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.$

Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n.$

Izvor: Međunarodna matematička olimpijada, shortlist 1988

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IMO Shortlist 1988 problem 1