Well I can't add much to the Perl solutions given by YuckFoo and particle. Heck, I'm having fun just working through them to understand the code and learn some more.
I do find Triangle Numbers interesting though. Here's a bit more about them on MathWorld. Extend the triangles into the third dimension and you get Tetrahedral Numbers. I wonder if a similar statement could be made with tetrahedral numbers.
Owl looked at him, and wondered whether to push him off the tree; but, feeling that he could always do it afterwards, he tried once more to find out what they were talking about.