Can we write each integer $n>12$ as $x+y+z$ with $T_x+T_y+T_z$ a triangular number?

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!

• I feel that my questions are usually not welcome here, and perhaps I should quit from MathOverflow. I don't know why this question receives so many downvotes. The problem should be at research level in my opinion. Can anybody tell me the reasons for downvotes on this one? – Zhi-Wei Sun Mar 11 at 7:46
• I think that some, not all, of your questions have varying measures of the following issues: 1) not aesthetically pleasing; 2) not naturally arising from other research fields; 3) deep into a land where proofs are currently hopeless and only probabilistic and numerical checks can be made; 4) the frequency of new posts, despite reservations already expressed by some in comments/answers. I warmly wish that you won't quit MO, but I would suggest stricter filtering of which problems you chose to submit. (Interested people can follow links to your site for more problems.) – Yaakov Baruch Mar 11 at 10:07
• At the level where you are working, "research level" isn't really a consideration. I agree with what Yaakov said, and would add that I think we all need to consider, before posting a question: do we expect that somebody on MO will be able to answer this? If the answer is 'no', then the question should be reconsidered (and maybe replaced by a subsidiary question that looks more promising). – Todd Trimble Mar 11 at 11:40
• I heard Erd\H os had two rules for problem sessions: 1. Write your name on the board, 2. don't ask for the Riemann hypothesis. In my opinion the best questions are questions where either you have the feeling that someone solved this or a similar problem somewhere, but you cannot find it, or where you ask about a topic quite different from your expertise, so you can assume that someone with another background can easily answer these questions. This does not mean that this question is bad, and I would neither downvote it nor vote to close it, but I would also not vote up. – Jan-Christoph Schlage-Puchta Mar 11 at 11:55
• If a problem can be solved without great efforts then I may simply ask my graduate students to study it, why should I post it to MathOverflow? This is my puzzle. People here usually have levels much higher than graduate students, but really challenging questions are not welcome here. Recall that Galois could solve the big problem on algebraic equations at his time. – Zhi-Wei Sun Mar 11 at 12:09