Upper Bounds for Stern's Diatomic Sequence and Related Sequences

Abstract

Let $(s_2(n){)}_{n=0}^{\infty }$ denote Stern's diatomic sequence. For $n\ge 2$, 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 $b,n\ge 2$, 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 $\underset{n\to \infty }{\mathrm{limsup}}\frac{s_b(n)}{n^{{\log }_b\ddot{I}•}}=\frac{(b^2-1{)}^{{\log }_b\ddot{I}•}}{\sqrt{5}}$.

61 pages, 4 figures