On the Orthogonality of Generalized Pattern Sequences

TL;DR

This paper introduces generalized pattern sequences that count multiple pattern occurrences in various base expansions and analyzes their partial sums, extending classical binary pattern studies to broader contexts.

Contribution

It presents a new framework for generalized pattern sequences in arbitrary bases and investigates their partial sum properties, expanding prior binary-focused research.

Findings

Analysis of partial sums for generalized pattern sequences

Extension of pattern counting to multiple patterns and bases

Foundational results for future studies in pattern sequence behavior

Abstract

The partial sums of integer sequences that count the occurrences of a specific pattern in the binary expansion of positive integers have been investigated by different authors since the 1950s. In this note, we introduce generalized pattern sequences, which count the occurrences of a finite number of different patterns in the expansion of positive integers in any integer base, and analyze their partial sums.