The $\textit{taxicab distance}$ between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane is given by \[|x_1 - x_2| + |y_1 - y_2|.\]
For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
$\textbf{(A)} \, 441 \qquad\textbf{(B)} \, 761 \qquad\textbf{(C)} \, 841 \qquad\textbf{(D)} \, 921  \qquad\textbf{(E)} \, 924$
Let us consider the number of points for a certain $x$-coordinate. For any $x$, the viable points are in the range $[-20 + |x|, 20 - |x|]$. This means that our total sum is equal to 
\begin{align*} 1 + 3 + 5 + \cdots + 41 + 39 + 37 + \cdots + 1 &= (1 + 3 + 5 + \cdots + 39) + (1 + 3 + 5 + \cdots + 41) \\ & = 20^2 + 21^2 \\ & = 29^2 \\ &= \boxed{\textbf{(C)} \, 841}. \end{align*}
~mathboy100