I'm trying to prove there are infinitely many positive integers that are not the sum of two triangular numbers. I know that a positive integer $n$ is called a triangular number if $n=\frac{1}{2}k(k+1)$ for some positive integer $k$. Here's what I have so far:
Proof: Let $T_n$ be the n'th triangle number where $n=1,2,3,...$. By the definition of triangular number, $T_n=\frac{n(n+1)}{2}$. Let $n=9k+r$, where $k,r$ are integers and $0\leq r < 9$. We have that
$T_n=\frac{(9k+r)(9k+r+1)}{2}=\frac{9k(9k+r+1)+9kr}{2}+\frac{r(r+1)}{2}$
At this point, I am stuck. I'm not sure where to go from here or even if I am on the right track. I think I may need to show that $\frac{9k(9k+r+1)+9kr}{2}$ is an integer divisible by 9 and need to give some explanation on the possible remainders, but I'm unsure if that is correct.
Any help or feedback is appreciated.