CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 10.1155/2016/4361582 4361582 Research Article A Symmetric Algorithm for Golden Ratio in Hyper-Horadam Numbers http://orcid.org/0000-0002-6356-6592 Bahşi Mustafa 1 http://orcid.org/0000-0003-4085-277X Solak Süleyman 2 Hu Qinghua 1 Education Faculty Aksaray University 68100 Aksaray Turkey aksaray.edu.tr 2 A. K. Education Faculty Konya N. E. University 42090 Konya Turkey konya.edu.tr 2016 1082016 2016 03 04 2016 17 06 2016 17 07 2016 1082016 2016 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.

We study some ratios related to hyper-Horadam numbers such as W n r / W n - 1 r while n by using a symmetric algorithm obtained by the recurrence relation a n k = a n k - 1 + a n - 1 k , where W n r is the n th hyper-Horadam number. Also, we give some special cases of these ratios such as the golden ratio and silver ratio.

1. Introduction

The Fibonacci numbers are defined by the second-order linear recurrence relation F n + 1 = F n + F n - 1 n 1 with the initial conditions F 0 = 0 and F 1 = 1 . Similarly, the Lucas numbers are defined by L n + 1 = L n + L n - 1 n 1 with the initial conditions L 0 = 2 and L 1 = 1 . The Fibonacci sequence can be generalized as the second-order linear recurrence W n ( a , b ; p , q ) , or briefly W n , defined by (1) W n + 1 = p W n + q W n - 1 , where n 1 , W 0 = a , and W 1 = b . This number sequence was introduced by Horadam . The characteristic equation of W n is (2) t 2 - p t - q = 0 . The roots of (2) are α = p + p 2 + 4 q / 2 and β = p - p 2 + 4 q / 2 . We think of α and β as being real, though this need not be so; that is, p 2 + 4 q 0 . The Binet formula for W n is (3) W n = A α n + B β n , where (4) A = b - a β p 2 + 4 q , B = a α - b p 2 + 4 q .

Some of the special cases of Horadam number W n are as follows: (5) the  Fibonacci  number   F n = W n 0,1 ; 1,1 , the  Lucas  number   L n = W n 2,1 ; 1,1 , the  Pell  number   P n = W n 0,1 ; 2,1 .

From (3)–(5) it follows that (6) P n = 2 4 α n - β n , F n = α 1 n - β 1 n 5 , L n = α 1 n + β 1 n , where (7) α 1 = 1 + 5 2 , β 1 = 1 - 5 2 ,

that is, α 1 and β 1 are the roots of (8) t 2 - t - 1 = 0 .

For the ratio W n + 1 / W n , (3) and (4) follow that (9) lim n W n + 1 W n = τ = α = p + p 2 + 4 q 2 if   p 0 β = p - p 2 + 4 q 2 if   p < 0 ; That is, τ is root of (2).

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: (10) lim n F n F n - 1 = 1 + 5 2 = ϕ golden  ratio , lim n L n L n - 1 = 1 + 5 2 = ϕ golden  ratio , lim n P n P n - 1 = 1 + 2 = Ψ silver  ratio .

The Euler-Seidel algorithm and its analogs are useful to study some recurrence relations and identities for some numbers and polynomials . Let a n and a n be two real initial sequences. Then the infinite matrix, which is called symmetric infinite matrix in , with entries a n k corresponding to these sequences is determined recursively by the following formulas: (11) a n 0 = a n , a 0 n = a n n 0 , a n k = a n k - 1 + a n - 1 k n 1 , k 1 . From (11), we can write (12) a n k = s = 0 s a s k - 1 . There are some applications of sequence (11) and its generalization . For example, Bahşi et al.  introduced the concepts as “hyper-Horadam” numbers and “generalized hyper-Horadam” numbers: (13) W n r = s = 0 n W s r - 1 = W n - 1 r + W n r - 1 w i t h W n 0 = W n , W 0 n = W 0 = a , W n r u , v = u W n r - 1 + v W n - 1 r , where u and v are two nonzero real parameters, W n 0 ( u , v ) = W n ( a , b ; p , q ) = W n and W 0 n ( u , v ) = W 0 ( a , b ; p , q ) = a , and W n is the n th Horadam number. Some of the special cases of hyper-Horadam number W n r are as follows:

If W n 0 = F n = W n ( 0,1 ; 1,1 ) and W 0 n = W 0 = F 0 = 0 , then W n r is the hyper-Fibonacci number; that is, W n r = F n r .

If W n 0 = L n = W n ( 2,1 ; 1,1 ) and W 0 n = W 0 = L 0 = 2 , then W n r is the hyper-Lucas number; that is, W n r = L n r .

If W n 0 = P n = W n ( 0,1 ; 2,1 ) and W 0 n = W 0 = P 0 = 0 , then W n r is the hyper-Pell number; that is, W n r = P n r .

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 W n r / W n - 1 r while n by using a symmetric algorithm obtained by the recurrence relation a n k = a n k - 1 + a n - 1 k .

2. Main Results Theorem 1.

Let the sequence a n k be as in (11). If lim n a n / a n - 1 = l , then for k 0 (14) lim n a n k a n - 1 k = l , where a 0 n = a and a is any real number.

Proof.

We use the principle of the mathematical induction on k . It is clear that the result is true for k = 0 ; that is, (15) lim n a n 0 a n - 1 0 = lim n a n a n - 1 = l . Let us assume that it is true for k - 1 ; that is, (16) l i m n a n k - 1 a n - 1 k - 1 = l . Then (17) l = lim n a n k - 1 a n - 1 k - 1 = lim n a 0 k - 1 + a 1 k - 1 + a 2 k - 1 + + a n k - 1 / n + 1 lim n a 0 k - 1 + a 1 k - 1 + a 2 k - 1 + + a n - 1 k - 1 / n = lim n a n k / n + 1 lim n a n - 1 k / n = lim n a n k a n - 1 k . That is, the result is true for k . 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 τ be as in (9). If r 0 , then, (18) lim n W n r W n - 1 r = τ .

Proof.

Since (19) lim n W n 0 W n - 1 0 = lim n W n W n - 1 = τ the proof is trivial from Theorem 1 if we select a n 0 = W n 0 = W n , a 0 n = W 0 = a , and a n r = W n r .

Theorem 3.

Let τ be as in (9). If r 1 , then, (20) i lim n W n r - 1 W n - 1 r = τ - 1 , i i lim n W n - 1 r W n r - 1 = 1 τ - 1 , i i i lim n W n r W n r - 1 = τ τ - 1 .

Proof.

(i) From Corollary 2, we have (21) lim n W n r W n - 1 r = τ . Then, (22) lim n W n r - 1 W n - 1 r = lim n W n r - W n - 1 r W n - 1 r = lim n W n r W n - 1 r - 1 = lim n W n r W n - 1 r - lim n 1 = τ - 1 .

(ii) From (i) (23) lim n W n - 1 r W n r - 1 = 1 lim n W n r - 1 / W n - 1 r = 1 τ - 1 .

(iii) Since lim n W n - 1 r / W n r - 1 = 1 / τ - 1 (from (ii)), we have (24) lim n W n r W n r - 1 = lim n W n r - 1 + W n - 1 r W n r - 1 = lim n 1 + W n - 1 r W n r - 1 = 1 + lim n W n - 1 r W n r - 1 = 1 + 1 τ - 1 = τ τ - 1 .

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 ϕ =    1 + 5 / 2 are as follows: (25) (i) lim n F n r F n - 1 r = lim n L n r L n - 1 r = ϕ , (ii) lim n F n r - 1 F n - 1 r = lim n L n r - 1 L n - 1 r = ϕ - 1 , (iii) lim n F n - 1 r F n r - 1 = lim n L n - 1 r L n r - 1 = ϕ , (iv) lim n F n r F n r - 1 = lim n L n r L n r - 1 = 1 + ϕ .

(2) The relationships between hyper-Pell numbers and silver ratio Ψ = 1 + 2 are as follows: (26) (i) lim n P n r P n - 1 r = Ψ , (27) (ii) l i m n P n r - 1 P n - 1 r = Ψ - 1 , (28) (iii) lim n P n - 1 r P n r - 1 = Ψ - 1 2 , (29) (iv) lim n P n r P n r - 1 = Ψ + 1 2 .

Competing Interests

The authors declare that they have no competing interests.

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