Scaled chance three uniform integers sum to N

For a positive integer $N$, choose $X_1,X_2,X_3$ independently and uniformly from $\{0,1,\dots,N\}$, and let $p_N$ be the probability that $X_1+X_2+X_3=N$. Determine $\lim_{N\to\infty}N\,p_N$ (enter $-1$ if the limit does not exist).

$1/2$. Solutions of $x_1+x_2+x_3=N$ in non-negative integers number $\binom{N+2}{2}$, and all lie in range, so $p_N=\tfrac{(N+2)(N+1)}{2(N+1)^3}$. Multiplying by $N$ and letting $N\to\infty$ gives $\tfrac12$.