# A Symmetric Algorithm for Golden Ratio in Hyper-Horadam Numbers

Authors: Mustafa Bahşi; Süleyman Solak
Journal: Chinese Journal of Mathematics
 (2016)

---

## Abstract

We study some ratios related to hyper-Horadam numbers such asW
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 nth hyper-Horadam number. Also, we give some special cases of these ratios such as the golden ratio and silver ratio.


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## Body

## 1. Introduction

The Fibonacci numbers are defined by the second-order linear recurrence relationF
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 [1]. 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 numberW
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 ratioW
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 [2–5]. Let a
n and a
n be two real initial sequences. Then the infinite matrix, which is called symmetric infinite matrix in [4], 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 [2–6]. For example, Bahşi et al. [6] 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 nth Horadam number. Some of the special cases of hyper-Horadam number W
n
r are as follows:(i)
IfW
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.
(ii)
IfW
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.
(iii)
IfW
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 ratioW
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 sequencea
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 onk. 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 Theorem1, we have the next corollary for the hyper-Horadam numbers.Corollary 2.
Letτ 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
=
τthe 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.
Letτ 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 Corollary2, 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) Sincelim
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 ratioϕ
=
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 ratioΨ
=
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
.



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*Source: 4361582-2016-08-10.xml*