A Note on Iterated Beatty Sequences

Mohsen Khani; Ali N. Valizadeh; Afshin Zarei

number theory

A Note on Iterated Beatty Sequences

arXiv:2607.12817

summary

The paper provides an elementary number‑theoretic criterion that exactly characterizes when a natural number belongs to the nth iterate of a Beatty sequence with irrational modulus α > (3+√5)/2.

Abstract

For any irrational number $I^\pm > \frac{3 + 5}{2} \approx 2.618$ and given a positive $n \in N$ , we use elementary number theory to introduce a necessary and sufficient condition for a natural number $x$ to be in the $n$ th iterate of the Beatty sequence of modulus $I^\pm$ .

10 pages

Topics & keywords

#beatty sequences #iterated sequences #irrational numbers #elementary number theory #membership criteriaBeatty sequenceiteratenecessary and sufficient conditionirrational αnatural numbers