A monotonicity property for generalized Fibonacci sequences

Year, pages: 2017, 585 - 592

Given $k ≥ 2$, let $a_n$ be the sequence defined by the recurrence $a_n=α₁a_n-1+…+α_ka_n-k$ for $n ≥ k$, with initial values $a₀=a₁=…=a_k-2=0$ and $a_k-1=1$. We show under a couple of assumptions concerning the constants $α_i$ that the ratio $\frac{\sqrt[n]{a_n}}{\sqrt[n-1]{a_n-1}}$ 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 $α_i$ are unity or when all of the $α_i$ 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.

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

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