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A monotonicity property for generalized Fibonacci sequences

In: Mathematica Slovaca, vol. 67, no. 3
Toufik Mansour - Mark Shattuck
Detaily:
Rok, strany: 2017, 585 - 592
Kľúčové slová:
monotonicity, log-concavity, $k$-Fibonacci numbers
O článku:
Given $k ≥ 2$, let $an$ be the sequence defined by the recurrence $an1an-1+…+αkan-k$ for $n ≥ k$, with initial values $a0=a1=…=ak-2=0$ and $ak-1=1$. We show under a couple of assumptions concerning the constants $αi$ that the ratio $\frac{\sqrt[n]{an}}{\sqrt[n-1]{an-1}}$ is strictly decreasing for all $n ≥ N$, for some $N$ depending on the sequence, and has limit $1$. In particular, this holds in the cases when all of the $αi$ are unity or when all of the $αi$ are zero except for the first and last, which are unity. Furthermore, when $k=3$ or $k=4$, it is shown that one may take $N$ to be an integer less than $12$ in each of these cases.
Ako citovať:
ISO 690:
Mansour, T., Shattuck, M. 2017. A monotonicity property for generalized Fibonacci sequences. In Mathematica Slovaca, vol. 67, no.3, pp. 585-592. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0292

APA:
Mansour, T., Shattuck, M. (2017). A monotonicity property for generalized Fibonacci sequences. Mathematica Slovaca, 67(3), 585-592. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0292
O vydaní:
Publikované: 27. 6. 2017