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Nice sequence

Source: Yugoslavia National Olympiad 2008

April 22, 2008
inductionquadraticsnumber theoryalgebra proposedalgebra

Problem Statement

The sequence (an)n≥1 (a_n)_{n\ge 1} is defined by a_1 \equal{} 3, a_2 \equal{} 11 and a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}, for n≥3 n \ge 3. Prove that each term of this sequence is of the form a^2 \plus{} 2b^2 for some natural numbers a a and b b.
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