Not only is this
true, but any
odd number can be represented as the difference of two
consecutive squares.
This is intimately related with basic calculus. The derivative (that is, rate of change) of a quadratic function like x^2 will change linearly as one moves along the function. That is, the rate of change at a given integer x-value along x^2 is two plus the rate of change at the previous integer x-value.
This trend is the same for higher powers of x. The rate of change of x^3 is proportional to x^2, and the rate of change of x^4 is proportional to x^3, etc.
BTW, I am one of those math nuts.