Putnam 1966 A1

A-1. Let $f(n)$ be the sum of the first $n$ terms of the sequence $0,1,1,2,2,3,3,4, \cdots$, where the $n$th term is given by $$a_{n}=\left\{\begin{array}{cc} n / 2 & \text { if } n \text { is even } \\ (n-1) / 2 & \text { if } n \text { is odd } \end{array}\right.$$

Show that if $x$ and $y$ are positive integers and $x>y$ then $x y=f(x+y)-f(x-y)$.