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arXiv · 1503.05305

On Generalized Fibonacci Numbers

Jerico B. Bacani·Julius Fergy T. Rabago
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Abstract

We provide a formula for the $n^{th}$ term of the $k$-generalized Fibonacci-like number sequence using the $k$-generalized Fibonacci number or $k$-nacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation $x + x^{-k} = 2$. We then extend our results to $k$-generalized Horadam ($k$GH) and $k$-generalized Horadam-like ($k$GHL) numbers. In dealing with the limit of the ratio of successive terms of $k$GH and $k$GHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for $k$-periodic $k$-nary Fibonacci sequence can also be derived.

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Jerico B. Bacani, Julius Fergy T. Rabago. 2015-03-18. On Generalized Fibonacci Numbers. https://doi.org/10.12988/ams.2015.5163

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