Let's determine the integer solutions of the equation

$$
3^{n}=2 n^{2}+1
$$
Solution. The expression $2 n^{2}+1$ has its minimum at $n=0$, where the value on the right side is 1. It immediately follows that in the case of $3^{n}2$, the current value grows more slowly than twice the actual value, so there are no more solutions to the equation in this domain.

The solutions to the equation: $n=0, n=1, n=2$.