Upper Bounds for Stern's Diatomic Sequence and Related Sequences

TL;DR

This paper develops methods to compute and establish upper bounds for Stern's diatomic sequence and related sequences, revealing asymptotic growth rates and improving existing bounds for the case b=2.

Contribution

It introduces a transfer-matrix approach to calculate sequence values and derives new upper bounds, including asymptotic growth rates, for these combinatorial sequences.

Findings

Improved upper bounds for Stern's sequence when b=2.

Established asymptotic growth rate of s_b(n) as n approaches infinity.

Derived a limit superior formula for the ratio s_b(n)/n^{log_b φ}.

Abstract

Let $(s_2(n))_{n=0}^\infty$ denote Stern's diatomic sequence. For $n\geq 2$, we may view $s_2(n)$ as the number of partitions of $n-1$ into powers of $2$ with each part occurring at most twice. More generally, for integers $b,n\geq 2$, let $s_b(n)$ denote the number of partitions of $n-1$ into powers of $b$ with each part occurring at most $b$ times. Using this combinatorial interpretation of the sequences $s_b(n)$, we use the transfer-matrix method to develop a means of calculating $s_b(n)$ for certain values of $n$. This then allows us to derive upper bounds for $s_b(n)$ for certain values of $n$. In the special case $b=2$, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$.