The average of an infinite sequence

Question

I ran into the following problem when I did scientific research.

Consider an infinite sequence $\{a_{i}\}$ for $p>u>0$. $a_{1}=0$. If $a_{i}\geq p$, $a_{i+1}=a_{i}-p$; otherwise, $a_{i+1}=a_{i}+u$. What is the average of this sequence?

I did some numerical experimentation and felt that the general formula for the average seems very complicated: it is neither monotone nor continuous in $(p,u)$.

Answer 1

By induction, $a_i=(p+u)\{i\frac{u}{p+u}\}$, where $\{x\}$ is a fractional part of $x$. Thus, by Weyl equidistribution theorem, if $p/u$ is irrational, the average equals $(p+u)/2$, otherwise $u/(p+u)=a/b$ for coprime positive integers $a, b$, and the average equals $(p+u) \frac{b-1}{2b}$.