SUMMARY
Let (s2(n))8n=0(s_2(n))_{n=0}^\infty denote Stern's diatomic sequence. For n=2n\geq 2, we may view s2(n)s_2(n) as the number of partitions of n-1n-1 into powers of 22 with each part occurring at most twice. More generally, for integers b,n=2b,n\geq 2, let sb(n)s_b(n) denote the number of partitions of n-1n-1 into powers of bb with each part occurring at most bb times. Using this combinatorial interpretation of the sequences sb(n)s_b(n), we use the transfer-matrix method to develop a means of calculating sb(n)s_b(n) for certain values of nn. This then allows us to derive upper bounds for sb(n)s_b(n) for certain values of nn. In the special case b=2b=2, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that lim sup\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}.