A Pattern Sequence Approach to Stern's Sequence

TL;DR

This paper introduces a novel pattern sequence method to analyze Stern's sequence, expressing it through binary word occurrences and their complements, providing a new perspective on its structure.

Contribution

It presents a new pattern sequence approach to represent Stern's sequence using binary word occurrences and their complements, offering fresh insights into its combinatorial properties.

Findings

Stern's sequence can be expressed via binary pattern counts.

The approach links sequence values to binary word complements.

Provides a new combinatorial interpretation of Stern's sequence.

Abstract

Let w be a binary string and let a_w (n) be the number of occurrences of the word w in the binary expansion of n. As usual we let s(n) denote the Stern sequence; that is, s(0)=0, s(1)=1, and for n >= 1, s(2n)=s(n) and s(2n+1)=s(n)+s(n+1). In this note, we show that s(n) = a_1 (n) + \sum_{w in 1 (0+1)*} s([w bar]) a_{w1} (n) where w bar denotes the complement of w (obtained by sending 0 to 1 and 1 to 0, and [w] denotes the integer specified by the word w interpreted in base 2.