A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$.
For example $F_6= \{0,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,1\}$.
The consecutive differences of $F_6$ are $S_6=\{1/6, 1/30, 1/20, 1/12, 1/15, 1/10, 1/10, 1/15, 1/12, 1/20, 1/30, 1/6\}$,
and the sum of squares of the elements of $S_6$, $\displaystyle T_6=\sum_{i\in S_6}i^2 = \frac{19}{180}$.
It appears that the sum of squares $T_n$ shrinks a bit faster than $O(ln(n)/n^2)$ but I cannot see how to prove it. I'd also be interested in considering higher powers than the square, and the differences between every other element, or every third, etc.