Recall that the triangular numbers are those natural numbers

$$T_n:=\frac{n(n+1)}2\quad \ (n\in\mathbb N=\{0,1,2,\ldots\}).$$
It is well known that each $n\in\mathbb N$ can be written as the sum of three triangular numbers.

**QUESTION.** Can we write each integer $n > 12$ as $x+y+z$ with $x,y,z$ positive integers such that the sum of the three triangular numbers $T_x,T_y,T_z$ is also a triangular number?

For example, $16 = 3+6+7$ with $T_3+T_6+T_7=T_{10}$.

I formulated this question in Oct. 2013 (cf. http://oeis.org/A230121) and checked it via Mathematica. It seems that the question should have a positive answer. Perhaps, some of you might be able to answer this question. Your comments are welcome!