<s>[INST] Prove this form $2^{n+2}-7a^2_{n}$ always is square numbers. Let sequence $\{a_{n}\}$ such $a_{1}=1,a_{2}=-1$,and 
$$a_{n}+a_{n-1}+2a_{n-2}=0,n\ge 3$$
show that 
$2^{n+2}-7a^2_{n}$ is square numbers.
such as
$$2^3-7=1,n=1$$
$$2^4-7=9=3^2,n=2$$
$a_{3}=-1$
$$2^{5}-7=25=5^2,n=3$$
$a_{4}=3$
$$2^6-7\cdot 9=1^2,n=4$$
 [/INST] We have always $2^{n+2}-7a^2_{n}=b_n^2$, where $b_1=-1$,  $b_2=-3$ and $b_n+b_{n-1}+2b_{n-2}=0$ (induction).
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