Theory of Arithmetico-Hierarchical Sequences

While classical arithmetic sequences are a cornerstone of mathemat
ics, their simplicity limits their ability to model systems with intertwined,
multi-layered rhythms. This article introduces a natural extension—arithmetico
hierarchical sequences—by incorporating a hierarchy of ratios {ri} and
steps {pi}, enabling the modeling of nested dynamic progressions through
an elegant stratified recurrence. Beyond their theoretical appeal, these
sequences offer promising cryptographic applications, as their multi-scale
complexity enhances security by making them difficult to reverse-engineer.
Additionally, their asymptotic behavior and algorithmic properties could
advance the modeling of complex systems in economics, biology, and com
puter science. We formally define these sequences, derive key theorems,
present efficient computation algorithms, and explore applications, aiming
to establish a foundation for future research in this versatile framework.