RMO is insane

For $n=1$ , $9+3+7=19$ is not a perfect square. For $n=2$ , $81+12+7=100$ is a perfect square.

For $n>2$ , we find that $3n^2+7<2\cdot 3^n+1$

$\implies 3^{2n}+3n^2+7<3^{2n}+2\cdot 3^n+1$

$\implies 3^{2n}+3n^2+7<(3^n+1)^2$

However, $(3^n)^2<3^{2n}+3n^2+7$ . This means that for $n>2$ , $3^{2n}+3n^2+7$ is strictly between two consecutive perfect squares, so it cannot be a square itself.

Hence the only solution is $n=\boxed{2}$ .