Questions tagged [farey-sequences]

Two fractions $\frac{p}{q}$, $\frac{p'}{q'}$ are said to be a Farey Pair if $pq'-p'q=\pm1$. Let $\frac{p}{q} < \frac{p'}{q'}$ be a Farey Pair, then $$ f:\left[\frac{0}{1}, \frac{1}{1}\right] \cap \mathbb{Q} \to \left[\frac{p}{q},…

This is a question about the Farey sequences (Fn) in the limit case as n approaches infinity. If we take the infinite case of the Farey sequences and treat all the points in this set as a linear array, is this set of points a 1D quasicrystal?

Given a number $j$, is there a way to determine that it is the length of a valid Farey sequence. The length is given by: $$|F_n| = \frac{1}{2}(n+3)n - \sum_{d=2}^n |F_\frac{n}{d}|$$ For example, 2, 7, 16875 all valid (lengths of sequence of orders…