We consider the time series \(x\) and \(w\) defined by \[x_n=\left\{\begin{array}{ll} n^2 & \mathrm{if\;}n\ge 0\\ 0 & \mathrm{if\;}n<0\end{array}\right.\] and \[w_n=\left\{\begin{array}{ll} \frac{n}{2} & \mathrm{if\;}|n|=1\\0 & \mathrm{otherwise}\end{array}\right.\] Calculate in the programming language that you selected at least 5 nonzero values of the cross-correlation \(x\otimes w\).
Then give an exact formula for the cross-correlation and prove with pen and paper that it is correct.