Question 2
The integer sequence $\{a_n\}_{n=0}^{+\infty}$ is defined by $a_0=1$, $a_1=3$, and
$$a_{n+2}=\begin{cases}a_{n+1}+9a_n,&n\text{ even},\\ 9a_{n+1}+5a_n,&n\text{ odd}.\end{cases}$$
Prove that $a_{1995}^2+a_{1996}^2+a_{1997}^2+a_{1998}^2+a_{1999}^2+a_{2000}^2$ is divisible by $20$, and that no term $a_{2m+1}$ is a perfect square.
$$a_{n+2}=\begin{cases}a_{n+1}+9a_n,&n\text{ even},\\ 9a_{n+1}+5a_n,&n\text{ odd}.\end{cases}$$
Prove that $a_{1995}^2+a_{1996}^2+a_{1997}^2+a_{1998}^2+a_{1999}^2+a_{2000}^2$ is divisible by $20$, and that no term $a_{2m+1}$ is a perfect square.