Sum of squares

In some sense, I think it happens more often than it "should" when you deal with messy systems in physics, for example. As a concrete example: Interactions between electric point charges (e.g. electrons) can be approximated in various ways with the Legendre polynomials. The n'th of these polynomials have a number of different series and contour integral representations that don't look particularly nice. There is one very nice representation of the n'th such polynomial, though: Take the polynomial x^2 - 1, raise it to the n'th power, then take the n'th derivative of the result (and multiply it by a simple constant). Pretty easy to remember. It's also a bit odd in that we don't encounter derivatives of an order higher than 2 (or maybe 3) very often IRL.

I feel this is a side effect of us having very beautiful mathematical language at our disposal. Our understanding of mathematics has come so far that even the most complicated aspects can be expressed in a simplistic form that is easy to understand. It's one of my favorite things about math.