Let $a_n$ be a sequence of integers defined as $$a_n= \left\lbrace \begin{array}{ll} (-1)^k & \text{if }n=\frac{k(3k\pm 1)}{2}\\ 1 & \text{if } n=0\\ 0 & \text{else } \end{array} \right.$$
How can I find a closed form for the discrete convolution $b_n = a_n \ast a_n$ defined by:
$$b_l = \sum_{n=0}^{l} a_n a_{l-n}$$
? This may be a simple and elementary problem, but I am struggling on getting a solution. Any help will be welcomed.
Thank you.