A lazy mathematician believes that $$a^2+b^2 = (a+b)^2.$$ If $a$ and $b$ are both integers from the interval $[-100, 100]$, find the number of ordered pairs $(a,b)$ that satisfy the equation above.
The equation implies that $2ab = 0$, hence either $a=0$ or $b=0$. This would result in the following pairs. When $a=0$ $$(0,-100), (0, -99), \dots, (0, 100).$$
When $b=0$ $$(-100, 0), (-99, 0), \dots, (100, 0)$$
Since we have twice the pair $(0, 0)$ we should subtract $1$ from the total counts. For me it seems that there's $200$ possible choices in each scenario. This would imply that the total would be $400-1 = 399$, but the correct answer was $401$. What am I missing in the countings?