I hope you will excuse my vague title. Let's define
$$\varsigma \ \colon\ \mathbb N\to \mathbb Z\ \colon\ k\mapsto (-1)^{\lfloor \frac{k-1}{2}\rfloor}$$ and $$S(m,n)=\displaystyle\sum_{k=m}^n k\cdot \varsigma(k)$$
Find every integer $n\ge 1$ such that $S(1,\lfloor n/2\rfloor)=S(\lfloor n/2\rfloor+1,n)$
Show that $-(n+1)\le \delta(n)\le 2(n+1)$, when $\delta(n)=S(1,\lfloor n/2\rfloor)-S(\lfloor n/2\rfloor+1,n)$