VMO 1984 — Division A Question 2

Let $u_1=1$, $u_2=2$, $u_{n+1}=3u_n-u_{n-1}$ for $n=2,3,\ldots$. Define $v_n=\sum_{k=1}^{n}\operatorname{arccot}u_i$ as printed. Determine $\lim_{n\to+\infty}v_n$.