Upper Bounds for Stern's Diatomic Sequence and Related Sequences

作者：Defant Colin^*

来源：ELECTRONIC JOURNAL OF COMBINATORICS, 2016, 23(4): P4.8.

DOI：10.37236/5342

• Stern sequence
• Fibonacci number
• Lucas number
• over-expansion
• transfer-matrix method

Let (s(2)(n))(n=0)(infinity) denote Stern's diatomic sequence. For n >= 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 >= 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 sb(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). In the special case b = 2, our bounds improve upon the current upper bounds for the Stern sequence. We then show that lim sup(n ->infinity) s(b)(n)/n(logb phi) = (b(2) - 1)(logb phi)/root 5.,

• 出版日期2016-10-14