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Upper Bounds for Stern's Diatomic Sequence and Related Sequences

By Colin Defant

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 $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$.Comment: 61 pages, 4 figure

Topics: Mathematics - Combinatorics, 05A17, 11B83, 11B39
Year: 2015
OAI identifier: oai:arXiv.org:1506.07824

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