For a geometric proof, see this page. For the Wikipedia article, see squared triangular number (two of my professors are in the references, even!). For a geometric generalization, see this paper. Sometimes the result you mention is called "Nichomachus' theorem". An algebraic generalization I wasn't aware of is Faulhaber's formula; that page contains several related identities including the very similar square pyramidal number formula and my first formula. For a rather interesting but somewhat advanced generalization dealing with Diophantine equation solutions, see this paper.
Reply With Quote