$\large a_n = \left\{\begin{matrix} a_{n-1}+2n+3, n \ge 0\\ 4, n = 0 \end{matrix}\right. \\ \\ \Rightarrow \sum_{1 \le k \le n} a_k-a_{k-1} = \sum_{1 \le k \le n}2k+3 \\ \Rightarrow a_n - a_0 = 3n+ 2\sum_{1 \le k \le n} k = 3n + 2\binom{n+1}{2}\\ \Rightarrow a_n = 4 + 3n+2\binom{n+1}{2}$