Three Series for the Generalized Golden Mean

2013

In this article we consider the infinite sums of reciprocal generalized Fibonacci numbers and the infinite sums of reciprocal generalized Fibonacci sums. Applying the floor function to the reciprocals of these sums, our results generalize some identities of Holliday and Komatsu and extend some results of Liu and Zhao. 1

Highly informative research paper

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.

International Journal of Quantitative Research and Modeling

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

Applied Mathematics and Computation, 2013

Article 10.5.8.] computed partial in…nite sums including reciprocal usual Fibonacci, Pell and generalized order-k Fibonacci numbers. In this paper we will present generalizations of earlier results by considering more generalized higher order recursive sequences with additional one coe¢ cient parameter.

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.

Fibonacci sequence is defined by the recurrence relation for with initial condition This sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. One of the generalizations of the Fibonacci sequence is the class of sequences generated by the recurrence relation with initial condition and a, b are positive integers. In this paper we express in simple explicit form and use it to derive the recursive formula for to compute its successor and predecessor. We also compute the value of &#39;generalized golden proportion&#39; for this sequence.

Applied Mathematics and Computation, 2015

Golden Ratio is defined by a proportion corresponding to the geometric mean. We introduce a generalized Golden Ratio as a fixed point of an operator defined by an arbitrary mean satisfying certain conditions. An algorithm for the evaluation of the generalized Golden Ratio is obtained using Banach's fixed point theorem. As applications we show that some problems in shape optimization correspond to some aesthetic standards introduced in the paper.

This book contains 50 papers from among the 95 papers presented at the Seventh International Conference on Fibonacci Numbers and Their Applications which was held at the Institut Fiir Mathematik, Technische Universitiit Graz, Steyrergasse 30, A-SOlO Graz, Austria, from July 15 to July 19, 1996. These papers have been selected after a careful review by well known referees in the field, and they range from elementary number theory to probability and statistics. The Fibonacci numbers and recurrence relations are their unifying bond. It is anticipated that this book, like its six predecessors, will be useful to research workers and graduate students interested in the Fibonacci numbers and their applications.

IJCSAM (International Journal of Computing Science and Applied Mathematics), 2023

The Fibonacci and Lucas sequences have been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. One of them is defined by the relation Bn = Bn−1 + Bn−2 n ≥ 2 with the initial condition B0 = 2s, B1 = s + 1 where s ∈ Z. In this paper, we consider the reciprocal sums of Bn and B 2 n , with an established result that also involve Bn.