Research Article | Open Access
A Symmetric Algorithm for Golden Ratio in Hyper-Horadam Numbers
We study some ratios related to hyper-Horadam numbers such as while by using a symmetric algorithm obtained by the recurrence relation , where is the th hyper-Horadam number. Also, we give some special cases of these ratios such as the golden ratio and silver ratio.
The Fibonacci numbers are defined by the second-order linear recurrence relation with the initial conditions and . Similarly, the Lucas numbers are defined by with the initial conditions and . The Fibonacci sequence can be generalized as the second-order linear recurrence , or briefly , defined by where , , and . This number sequence was introduced by Horadam . The characteristic equation of isThe roots of (2) are and . We think of and as being real, though this need not be so; that is, . The Binet formula for iswhere
Some of the special cases of Horadam number are as follows:
that is, and are the roots of
Over the past five centuries, the golden ratio has been very attractive for researchers because its occurrence is ubiquitous such as nature, art, architecture, and anatomy. From (9), we have the well-known golden ratio and silver ratio as follows:
The Euler-Seidel algorithm and its analogs are useful to study some recurrence relations and identities for some numbers and polynomials [2–5]. Let and be two real initial sequences. Then the infinite matrix, which is called symmetric infinite matrix in , with entries corresponding to these sequences is determined recursively by the following formulas:From (11), we can writeThere are some applications of sequence (11) and its generalization [2–6]. For example, Bahşi et al.  introduced the concepts as “hyper-Horadam” numbers and “generalized hyper-Horadam” numbers:where and are two nonzero real parameters, and , and is the th Horadam number. Some of the special cases of hyper-Horadam number are as follows:(i)If and , then is the hyper-Fibonacci number; that is, .(ii)If and , then is the hyper-Lucas number; that is, .(iii)If and , then is the hyper-Pell number; that is, .
The fundamental aim of this paper is to obtain relationships between special ratios such as the golden ratio, silver ratio, and hyper-numbers such as hyper-Fibonacci, hyper-Lucas, and hyper-Pell numbers. For this, we firstly investigate the ratio while by using a symmetric algorithm obtained by the recurrence relation
2. Main Results
Theorem 1. Let the sequence be as in (11). If , then for where and is any real number.
Proof. We use the principle of the mathematical induction on . It is clear that the result is true for ; that is, Let us assume that it is true for ; that is, Then That is, the result is true for . Thus the proof is completed.
As an application of the Theorem 1, we have the next corollary for the hyper-Horadam numbers.
Corollary 2. Let be as in (9). If , then,
Proof. Since the proof is trivial from Theorem 1 if we select , , and .
Theorem 3. Let be as in (9). If , then,
Proof. (i) From Corollary 2, we have Then, (ii) From (i) (iii) Since (from (ii)), we have
From these results we have some particular results for the relationships between hyper-Fibonacci, hyper-Lucas (hyper-Pell) numbers, and the golden (silver) ratio as follows:
(1) The relationships between hyper-Fibonacci (and Lucas) numbers and golden ratio are as follows:
(2) The relationships between hyper-Pell numbers and silver ratio are as follows:
The authors declare that they have no competing interests.
- A. F. Horadam, “Basic properties of a certain generalized sequence of numbers,” The Fibonacci Quarterly, vol. 3, pp. 161–176, 1965.
- N.-N. Cao and F.-Z. Zhao, “Some properties of hyperfibonacci and hyperlucas numbers,” Journal of Integer Sequences, vol. 13, article 10.8.8, 2010.
- A. Dil, V. Kurt, and M. Cenkci, “Algorithms for Bernoulli and allied polynomials,” Journal of Integer Sequences, vol. 10, no. 5, Article 07.5.4, 14 pages, 2007.
- A. Dil and I. Mezö, “A symmetric algorithm for hyperharmonic and Fibonacci numbers,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 942–951, 2008.
- D. Dumont, Matrices d. Euler. Seidel, Seminaire Lotharingien de Combinatorie, B05c, 1981, http://www.emis.de/journals/SLC/opapers/s05dumont.pdf.
- M. Bahşi, I. Mezö, and S. Solak, “A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers,” Annales Mathematicae et Informaticae, vol. 43, pp. 19–27, 2014.
Copyright © 2016 Mustafa Bahşi and Süleyman Solak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.