KöMaL 2016 B4766

The sequence $a_{1}, a_{2}, \dots$ is defined by the following recursion: $a_{1}=1$, $a_{2}=5$, $a_{3}=15$, and in the case of $n\ge 4$, $a_{n}=n^{2}+a_{n-1}+a_{n-2}-a_{n-3}$. $a)$ Calculate the sum $a_{1}+a_{2}+a_{3}+\dots +a_{2015}$. $b)$ Prove that $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+ \dots +\frac{1}{a_{2015}}<\frac{4}{3}$.