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On the Orthogonality of Generalized Pattern Sequences

Shuo Li

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.

On the Orthogonality of Generalized Pattern Sequences

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.
Paper Structure (3 sections, 5 theorems, 25 equations)

This paper contains 3 sections, 5 theorems, 25 equations.

Table of Contents

  1. Introduction, definitions and notation
  2. Window functions and $(a_{b,m,S}(n))_{n \in \mathbf{N}}$
  3. Application

Key Result

Proposition 1

Let $b,m$ be two integers larger than $1$ and let $S_1, S_2$ be two finite weighted subsets of $[\![b]\!]^*$. For any non-negative integer $n$, one has

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Example 6
  • Example 7
  • Example 8