Let $(a_n)_{n=1}^{\infty}$ be a real sequence such that $a_1=1$, $a_3=4$, and for every $n \geq 2$, $a_{n+1} + a_{n-1} = 2a_n + 1$. Find the value of $a_{2011}$.

Let's think step by step and output the final answer within \boxed{}.
user