5. Let $g_{3}(n)$ denote the number of $n$-partitions with exactly three parts, where exactly two parts are equal. Find $g_{3}(n)$.


5. $g_{3}(n)=\left\{\begin{array}{l}{\left[\frac{n-1}{2}\right], \text { if } 3 \nmid n ;} \\ {\left[\frac{n-1}{2}\right]-1, \text { if } 3 \mid n}\end{array}\right.$