I came up with this problem after discussion of taxicab geometry in math class... I thought it was a simple problem, but still pretty neat; however, I am as of yet unsure of whether my answer is correct, or logical.
Let $[X]$ be the area of region $X$, and region $S_n$ be represented by the equation $|x-n|+|y-n|=k-n$ for all $n=0,1,2,\ldots,k-1$. Now let region $R_n$ be the region between $S_n$ and $S_{n+1}$ and $L=\displaystyle\sum_{n=0}^{k-2}{[R_n]}$. Find the smallest positive integer $k$ such that $L > A$. ($A$ is any number you can plug in)
Can anyone else verify my result of $L=\frac{5k^2-k-4}{2}$?