AT-OEMO-REG 2007 Problem 3

Let $a$ be a positive real number and $n$ a non-negative integer. Determine $S-T$ , where $S= \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$ and $T= \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$