A Pattern Sequence Approach to Stern's Sequence

2011/04/30 by Michael James Coons, Michael Coons, Jeffrey Shallit +2
Computer Science · Mathematics · #Algorithms and Data Compression #Coding theory and cryptography #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.DM #math.NT #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1105.0086

openalex publication_date 2011/04/30 · arxiv created 2011/07/06 · arxiv updated 2011/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) = a₁ (n) + ∑_{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.

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• Binary Signed-Digit Integers, the Stern Diatomic Sequence and Stern Polynomials
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