Finite sum $\sum_{j=0}^{n-1} j^2$ How can I calculate this finite sum? Can someone help me?  

$$\sum_{j=0}^{n-1} j^2$$


I like this explanation:

Each row in the first triangle sums to $j^2$
So the sum of the rows is the number we seek.
Take that triangle and rotate left and right.  The sum is of the three trianges, then is 3 times the number we seek.  But when we sum it up, we get $(2n+1)$ in every entry in the triange or $(2n+1)$ times the number of entries.
$3\sum_\limits{j = 1}^n j^2 = (2n+1)\sum_\limits{j = 1}^n j$
