Upper Bounds for Stern's Diatomic Sequence and Related Sequences

Let $(s_2(n){)}_{n=0}^{\mathrm{\infty }}$ 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 $b,n\geq 2$,let$s_b(n)$denotethenumberofpartitionsof$n-1$intopowersof$b$witheachpartoccurringatmost$b$ 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 $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$.