A note on the Fibonacci sequence of order K and the multinomial coefficients (original) (raw)

On Generalized Fibonacci Numbers 1

2018

We provide a formula for the nth term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci 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 (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, 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. Mathematics Subject Classification: 11B39, 11B50.

Generalized Fibonacci sequences and their properties

arXiv (Cornell University), 2021

Let Fn(k) be the generalized Fibonacci number defined by (with Fi(k) abbreviated to Fi): Fn = Fn−1 + Fn−2 + • • • + F n−k , for n ≥ k, and the initial values (F0, F1, ..., F k−1). Let Bn(k, j) be Fn(k) with initial values given by Fj = 1 and, for i < j and j < i < k, Fi = 0. This paper shows that any Fn(k) can be expressed as the sum of Bn(k, j)s. This paper also expresses Bn(k, j) and Fn(k) as finite sums, derives some properties and evaluates their 2-adic order for a range of values of k, j and n and those of Bn(3, j) and Bn(4, j) for most values of j and n.

On Generalized Fibonacci Numbers

Applied Mathematical Sciences, 2015

We provide a formula for the n th term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci 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 (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, 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.

On linear recursive sequences with coefficients in arithmetic-geometric progressions

Applied mathematical sciences, 2015

We provide a formula for the n th term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci 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 (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, 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.