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 $x^2$.

$\sum_{k=0}^{x-1}2k=\left(0+\left(x-1\right)\right)+\left(1+\left(x-2\right)\right)+\cdots+\left(\left(x-1\right)+0\right)=x\left(x-1\right)=x^{2}-x$
hence:
$\sum_{k=0}^{x-1}(2k+1)=\sum_{k=0}^{x-1}2k+x=x^2$
