Undergraduate

Given the sequence $a _ 1 = 1 $ , $a _ 2 = 1 $ , and for $n geq 1 $ we

Given the sequence $a_1=1$, $a_2=1$, and for $n\geq1$ we have $a_{n +2}=a_{n +1}+2a_n$. Let $S(m)=a_1^2+ a_2^2+\cdots+a_m^2$. Find the value of $S(2022)\bmod{10000}$.

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