First things first. I've made this a seperate thread off my previous topic so that these topics don't conflict if I'd posted this to my previous question.
I believe any $k$ triangular numbers will have a unique sum, such that they can not be composed by any other $k$ triangular numbers. I believe my conjecture is right, here's my proof (other than experimenting);
Take two triangular numbers $(k=2)$ $a, b$ and $c, d$. Assuming the conjecture is false, the equation must prove;
$a(a+1)/2 + b(b+1)/2=c(c+1)/2 + d(d+1)/2$
which implies (after simplification) $a + b + a^2 + b^2 = c + d + c^2 + d^2$
This means that the natural numbers $a, b$ and $c, d$ must be equal to each other's squares, as well as their sum I'm not at all sure about this part...
Maybe that isn't even a valid proof, but if someone thinks the conjecture is right or wrong, please bother to give a proof why!
Important extra information: Fine, I've got all answers for k=2 uptil now, what if k were to be 3, 4 or greater? Are there possibilities then?