You're probably better at math than I am. I heard of this technique a long time ago; I don't remember where. I didn't know what it was called so I just googled "how to solve a sequence" and the page I linked above was the first result.
It does seem kind of magical. The math behind it is probably quite interesting. Try googling "method of common differences". If you find an explanation, let me know.
As for why it generates the FITs, I don't know. I just tried the technique on paper for the n-cubed example and it worked, so I implemented it for the question and got the right answer.
And note that for "project euler" (and similar) type of questions, there's often some kind of trick to simplify the algorithm. If you find yourself resorting to 3rd-party libraries or 300 lines of code or an hour of running time then you've probably missed it. At first problem 101 seemed extremely difficult until I remembered that trick.
