I found this odd relationship, $x^2 = \sum_\limits{k = 0}^{x-1} (2k + 1)$. I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.

Notice : $$\begin{align}(x + 1)^2 - x^2 &= x^2 + 2x + 1 - x^2 \\&= 2x + 1\end{align}$$
We take a summation on both sides and see that a lot of cancellation occurs on the LHS: 
$$\sum_{k = 0}^{x-1}\left((x+1)^2 - x^2\right) = \sum_{k = 0}^{x-1}(2x+1)\\
(x -1 + 1)^2 - 0^2 = \sum_{k = 0}^{x-1}(2x+1)\\
x^2 = \sum_{k = 0}^{x-1}(2x+1)$$
