Problem 44: Nice sequence

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Joined: Mar 10, 2015

by henderson, Jul 29, 2016, 10:31 AM

$\color{red}\bf{Problem \ 44}$ Let $k$ be an integer greater than $1.$ The sequence $\{a_n \}_{n\geq 1}$ is defined by
$a_0=4, a_1=a_2=(k^2-2)^2$ and $a_{n+1}=a_{n}a_{n-1}-2(a_{n}+a_{n-1})-a_{n-2}+8$ for $n\geq 2.$ Prove that $2+\sqrt{a_n}$ is a perfect square for all $n.$