# Dynamics and Solutions of a Fifth-Order Nonlinear Difference Equation

Authors: M. M. El-Dessoky; E. M. Elabbasy; Asim Asiri
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

The main objective of this paper is to study the behavior of the rational difference equation of the fifth-orderyn+1=αyn+βynyn-3/(Ayn-4+Byn-3), n=0,1,…, where α,β,A, and B are real numbers and the initial conditions y-4,y-3,y-2,y-1 and y0 are positive real numbers such that Ay-4+By-3≠0. Also, we obtain the solution of some special cases of this equation.


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## Body

## 1. Introduction

In recent years, there has been a great interest in studying the rational difference equations. These equations describe real life situations in stochastic time series, combinatorial analysis, electrical network, number theory, biology, genetics, probability theory, physics, ecology, statistical problems, and economics, for example [1–5]. It is so important to investigate the asymptotic behavior of solutions of a nonlinear difference equations and to discuss the boundedness, periodicity, and stability (local and global) of their equilibrium points; see [6–36] and references therein.Kalabušić et al. [6] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the difference equation (1)xn+1=βxn-l+δxn-kBxn-l+Dxn-k.In [7], Elabbasy et al. got the solution and the periodicity character of the recursive sequence (2)xn+1=axn-bxncxn-dxn-1.Cinar [8] found the solution of the difference equation (3)xn+1=axn-11+bxnxn-1.In [9] Obaid et al. studied the global stability, boundedness, and the periodicity of solutions of the rational difference equation (4)xn+1=axn+bxn-1+cxn-2+dxn-3αxn-1+βxn-2+γxn-3.Elsayed et al. [10] studied the dynamical analysis of rational difference equation (5)xn+1=αxn-3A+Bxn-1xn-3.Elabbasy et al. [11] obtained the global behavior of the solutions of the difference equation (6)xn+1=axn-lxn-kbxn-p+cxn-q.Aloqeili [12] investigated the dynamics of the difference equation (7)xn+1=xn-kA+xnxn-k.The aim of this paper is to study some qualitative behavior of the positive solutions of the difference equation(8)yn+1=αyn+βynyn-3Ayn-4+Byn-3,n=0,1,…,where α,β,A, and B are real numbers and the initial conditions y-4,y-3,y-2,y-1, and y0 are positive real numbers such that Ay-4+By-3≠0.


## 2. Preliminaries

LetI be some interval of real numbers and let (9)G:Il+1⟶I,be a continuously differentiable function. Then for every set of initial conditions y-l,y-l+1,…,y0∈I, the difference equation(10)yn+1=Gyn,yn-1,…,yn-l,n=0,1,…has a unique solution {yn}n=-l∞.Definition 1 (equilibrium point).
A pointy¯∈I is called an equilibrium point of the difference equation (10) if (11)y¯=Gy¯,y¯,…,y¯.That is, yn=y¯ for n≥0 is a solution of the difference equation (10) or, equivalently, y¯ is a fixed point of G.Definition 2 (stability).
Lety¯∈0,∞ be an equilibrium point of the difference equation (10). Then, we have the following:
(i) The equilibrium pointy¯ of the difference equation (10) is called locally stable if for every ϵ>0,there exists δ>0 such that for all y-l,y-l+1,…,y0∈I with (12)y-l-y¯+⋯+y-1-y¯+y0-y¯<δ,we have (13)yn-y¯<ϵfor  alln≥-l.
(ii) The equilibrium pointy¯ of the difference equation (10) is called locally asymptotically stable if y¯ is locally stable solution of (10) and there exists γ>0, such that, for all y-l,…,y-1,y0∈I with (14)y-l-y¯+⋯+y-1-y¯+y0-y¯<γ,we have (15)limn→∞yn=y¯.
(iii) The equilibrium pointy¯ of the difference equation (10) is called global attractor if for all y-l,…,y-1,y0∈I, we have (16)limn→∞⁡yn=y¯.
(iv) The equilibrium pointy¯ of the difference equation (10) is called globally asymptotically stable if y¯ is locally stable, and y¯ is also a global attractor of the difference equation (10).
(v) The equilibrium pointy¯ of the difference equation (10) is called unstable if y¯ is not locally stable.Definition 3 (periodicity).
A sequence{yn}n=-l∞ is said to be periodic with period p if yn+p=yn for all n≥-l. A sequence {yn}n=-l∞ is said to be periodic with prime period p if p is the smallest positive integer having this property.Definition 4 (Fibonacci sequence).
The sequence{Fi}i=0∞=1,1,2,3,5,8,13,21,, i.e., Fi=Fi-1+Fi-2,F-2=1,F-1=1, is called Fibonacci sequence.Definition 5.
The linearized equation of the difference equation (10) about the equilibrium y¯ is the linear difference equation(17)zn+1=∑i=0l∂Hy¯,y¯,…,y¯∂yn-izn-i.Now, assume that the characteristic equation associated with (17) is(18)pλ=p0λl+p1λl-1+⋯+pl-1λ+pl=0,where (19)pi=∂Hy¯,y¯,…,y¯∂yn-i.Theorem 6 (see [1]).
Assume thatpi∈R, i=1,2,…,l, and l is nonnegative integer. Then (20)∑i=1lpi<1is a sufficient condition for the asymptotic stability of the difference equation (21)yn+l+p1yn+l-1+⋯+plyn=0,n=0,1,…,Theorem 7 (see [2]).
Letg:[η,ξ]×[η,ξ]×[η,ξ]→[η,ξ] be a continuous function and [η,ξ] be an interval of real numbers. Consider the difference equation(22)yn+1=gyn,yn-3,yn-4.Assume that g satisfies the following conditions:
(i)g(x,y,z) is nondecreasing in x and y in [η,ξ] for all z∈[η,ξ] and nonincreasing in z∈[η,ξ] for all x and y in [η,ξ].
(2) If(m,M)∈[η,ξ]×[η,ξ] is a solution of the system (23)M=gM,M,mandm=gm,m,M,then (24)m=M.Then (22) has a unique equilibrium point y¯∈[η,ξ] and every solution of (22) converges to y¯.


## 3. Dynamics of (8)

In this section, we study the local stability, global stability of the solutions, and the boundedness of(25)yn+1=αyn+βynyn-3Ayn-4+Byn-3,whereα,β,A, and B are positive real numbers.


### 3.1. Local Stability of the Equilibrium Point

In this subsection, we study the local stability of the equilibrium point of (8).Equation (8) has a unique equilibrium point and is given by (26)y¯=αy¯+βy¯2Ay¯+By¯, or (27)1-αA+By¯2=βy¯2. If 1-αA+B≠β, then the only equilibrium point is y¯=0.Theorem 8.
Let(28)A+B21-α>β3A+B.Then the equilibrium point y¯ of (8) is locally asymptotically stable.Proof.
LetG:(0,∞)3→(0,∞) be a continuous function defined by (29)Gu1,u2,u3=αu1+βu1u2Au3+Bu2.
Therefore, it follows that(30)∂Gu1,u2,u3∂u1=α+βu2Au3+Bu2,∂Gu1,u2,u3∂u2=βAu1u3Au3+Bu22,∂Gu1,u2,u3∂u3=-βAu1u2Au3+Bu22.Then(31)∂Gy¯,y¯,y¯∂u1=α+βA+B,∂Gy¯,y¯,y¯∂u2=βAA+B2,∂Gy¯,y¯,y¯∂u3=-βAA+B2,
and the linearized equation of (8) about y¯=0 is (32)zn+1-α+βA+Bzn-βAA+B2zn-3+βAA+B2zn-4=0, It follows by Theorem 6 that (8) is asymptotically stable if and only if (33)α+βA+B+βAA+B2+βAA+B2<1, and so (34)α+βA+B+2βAA+B2<1, and, thus, (35)β3A+B<1-αA+B2.The proof is complete.Example 9.
Figure1 shows that the zero solution of the difference Equation (8) is local stability if α=0.4, β=0.03, A=0.1, and B=2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 1
Plot the behavior of the zero solution of (8).


### 3.2. Global Stability of the Equilibrium Point

In this subsection we study the global stability of the positive solutions of (8).Theorem 10.
The equilibrium pointy¯ of (8) is global stability if(36)B1-α≠β.Proof.
Letη and ξ be nonnegative real numbers and assume that g:[η,ξ]3→[η,ξ] is a function defined by (37)gv1,v2,v3=αv1+βv1v2Av3+Bv2.Then we can see that the function g(v1,v2,v3) is increasing in v1 and v2 and decreasing v3.
Assume that(m,M) is a solution of the system (38)M=gM,M,mandm=gm,m,M,
Then from (8), we see that(39)M=αM+βM2Am+BMandm=αm+βm2AM+Bm,and then (40)B1-αM2+A1-αMm=βM2,B1-αm2+A1-αMm=βm2. Subtracting these two equations, we obtain (41)B1-α-βM2-m2=0, and if B(1-α)-β≠0, then we see that M=m.
According to Theorem7 the equilibrium point y¯ is a global attractor of (8). The proof is complete.Example 11.
The zero solution of the difference Equation (8) is global stability if α=0.01, β=0.001, A=0.5, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1 (See Figure 2).Figure 2
Plot the behavior of the zero solution of (8) is global stable.


### 3.3. Existence of Boundedness and Unboundedness Solutions

Here we look at the boundedness and unboundedness solutions of (8).Theorem 12.
Every solution of (8) is bounded if(42)α+βB<1.Proof.
Let{yn}n=-4∞ be a solution of (8). It follows from (8) that (43)yn+1=αyn+βynyn-3Ayn-4+Byn-3,≤αyn+βynyn-3Byn-3=α+βByn. Then (44)yn+1≤ynfor  alln≥0. Then the sequence {yn}n=0∞ is decreasing and so is bounded from the above by M=maxy-4,y-3,y-2,y-1,y0.Theorem 13.
Every solution of (8) is unbounded if α>1.Proof.
Let{yn}n=-4∞ be a solution of (8). It follows from (8) that (45)yn+1=αyn+βynyn-3Ayn-4+Byn-3>αyn,for  alln≥1.We see that the right hand side can be written as follows: (46)xn+1=axn⟹xn=anx0, and this equation is unstable because a>1 and limn→∞⁡xn=∞. Then by using ratio test {yn}n=-4∞ is unbounded from above.Example 14.
Figure3 shows that (8) is bounded when α=0.15, β=0.1, A=0.1, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 3
Sketch the behavior of the solution of (8) is bounded.Example 15.
Figure4 shows that (8) is unbounded when α=1.2, β=0.1, A=0.1, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 4
Sketch the behavior of the solution of (8) is unbounded.


## 3.1. Local Stability of the Equilibrium Point

In this subsection, we study the local stability of the equilibrium point of (8).Equation (8) has a unique equilibrium point and is given by (26)y¯=αy¯+βy¯2Ay¯+By¯, or (27)1-αA+By¯2=βy¯2. If 1-αA+B≠β, then the only equilibrium point is y¯=0.Theorem 8.
Let(28)A+B21-α>β3A+B.Then the equilibrium point y¯ of (8) is locally asymptotically stable.Proof.
LetG:(0,∞)3→(0,∞) be a continuous function defined by (29)Gu1,u2,u3=αu1+βu1u2Au3+Bu2.
Therefore, it follows that(30)∂Gu1,u2,u3∂u1=α+βu2Au3+Bu2,∂Gu1,u2,u3∂u2=βAu1u3Au3+Bu22,∂Gu1,u2,u3∂u3=-βAu1u2Au3+Bu22.Then(31)∂Gy¯,y¯,y¯∂u1=α+βA+B,∂Gy¯,y¯,y¯∂u2=βAA+B2,∂Gy¯,y¯,y¯∂u3=-βAA+B2,
and the linearized equation of (8) about y¯=0 is (32)zn+1-α+βA+Bzn-βAA+B2zn-3+βAA+B2zn-4=0, It follows by Theorem 6 that (8) is asymptotically stable if and only if (33)α+βA+B+βAA+B2+βAA+B2<1, and so (34)α+βA+B+2βAA+B2<1, and, thus, (35)β3A+B<1-αA+B2.The proof is complete.Example 9.
Figure1 shows that the zero solution of the difference Equation (8) is local stability if α=0.4, β=0.03, A=0.1, and B=2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 1
Plot the behavior of the zero solution of (8).


## 3.2. Global Stability of the Equilibrium Point

In this subsection we study the global stability of the positive solutions of (8).Theorem 10.
The equilibrium pointy¯ of (8) is global stability if(36)B1-α≠β.Proof.
Letη and ξ be nonnegative real numbers and assume that g:[η,ξ]3→[η,ξ] is a function defined by (37)gv1,v2,v3=αv1+βv1v2Av3+Bv2.Then we can see that the function g(v1,v2,v3) is increasing in v1 and v2 and decreasing v3.
Assume that(m,M) is a solution of the system (38)M=gM,M,mandm=gm,m,M,
Then from (8), we see that(39)M=αM+βM2Am+BMandm=αm+βm2AM+Bm,and then (40)B1-αM2+A1-αMm=βM2,B1-αm2+A1-αMm=βm2. Subtracting these two equations, we obtain (41)B1-α-βM2-m2=0, and if B(1-α)-β≠0, then we see that M=m.
According to Theorem7 the equilibrium point y¯ is a global attractor of (8). The proof is complete.Example 11.
The zero solution of the difference Equation (8) is global stability if α=0.01, β=0.001, A=0.5, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1 (See Figure 2).Figure 2
Plot the behavior of the zero solution of (8) is global stable.


## 3.3. Existence of Boundedness and Unboundedness Solutions

Here we look at the boundedness and unboundedness solutions of (8).Theorem 12.
Every solution of (8) is bounded if(42)α+βB<1.Proof.
Let{yn}n=-4∞ be a solution of (8). It follows from (8) that (43)yn+1=αyn+βynyn-3Ayn-4+Byn-3,≤αyn+βynyn-3Byn-3=α+βByn. Then (44)yn+1≤ynfor  alln≥0. Then the sequence {yn}n=0∞ is decreasing and so is bounded from the above by M=maxy-4,y-3,y-2,y-1,y0.Theorem 13.
Every solution of (8) is unbounded if α>1.Proof.
Let{yn}n=-4∞ be a solution of (8). It follows from (8) that (45)yn+1=αyn+βynyn-3Ayn-4+Byn-3>αyn,for  alln≥1.We see that the right hand side can be written as follows: (46)xn+1=axn⟹xn=anx0, and this equation is unstable because a>1 and limn→∞⁡xn=∞. Then by using ratio test {yn}n=-4∞ is unbounded from above.Example 14.
Figure3 shows that (8) is bounded when α=0.15, β=0.1, A=0.1, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 3
Sketch the behavior of the solution of (8) is bounded.Example 15.
Figure4 shows that (8) is unbounded when α=1.2, β=0.1, A=0.1, and B=0.2 and the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 4
Sketch the behavior of the solution of (8) is unbounded.


## 4. Special Cases of (8)

In this section we investigate the following special case:(47)yn+1=αyn+βynyn-3Ayn-4+Byn-3,whereα, β, A, and B are integrals numbers.


### 4.1. First Case Whenα=β=A=B=1

Theorem 16.
Suppose that{yn}n=-4∞ is a solution of rational difference equation(48)yn+1=yn+ynyn-3yn-4+yn-3.Then, for n=0,1,2,…, we see that (49)y4n-3=d∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-2=c∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,where y-4=e,y-3=d, y-2=c, y-1=b, y0=a, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,(50)y4n-8=a∏i=0n-3aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-7=d∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-6=c∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-5=b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-4=a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1.
From (48), we see that (51)
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n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
d
a
b
b
c
c
d
2
a
+
b
a
+
b
2
b
+
c
b
+
c
2
c
+
d
c
+
d
5
a
+
3
b
3
a
+
b
5
b
+
3
c
3
b
+
c
5
c
+
3
d
3
c
+
d
⋯
a
F
2
n
-
3
+
b
F
2
n
-
4
a
F
2
n
-
4
+
b
F
2
n
-
5
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
a
2
a
+
b
a
+
b
2
b
+
c
b
+
c
2
c
+
d
c
+
d
5
a
+
3
b
3
a
+
b
5
b
+
3
c
3
b
+
c
5
c
+
3
d
3
c
+
d
⋯
a
F
2
n
-
3
+
b
F
2
n
-
4
a
F
2
n
-
4
+
b
F
2
n
-
5
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
a
b
b
c
c
d
2
a
+
b
a
+
b
2
b
+
c
b
+
c
2
c
+
d
c
+
d
5
a
+
3
b
3
a
+
b
5
b
+
3
c
3
b
+
c
5
c
+
3
d
3
c
+
d
⋯
a
F
2
n
-
3
+
b
F
2
n
-
4
a
F
2
n
-
4
+
b
F
2
n
-
5
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
d
a
b
b
c
c
d
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
a
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
a
b
b
c
c
d
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
a
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
a
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
/
d
F
2
n
-
2
+
e
F
2
n
-
3
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
/
d
F
2
n
-
2
+
e
F
2
n
-
3
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
-
2
+
e
F
2
n
-
3
+
d
F
2
n
-
1
+
e
F
2
n
-
2
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
+
e
F
2
n
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
F
2
n
+
e
F
2
n
-
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
+
e
F
2
n
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
F
2
n
+
1
+
e
F
2
n
d
F
2
n
+
e
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
.
(52)
y
4
n
-
2
=
y
4
n
-
3
+
y
4
n
-
3
y
4
n
-
6
y
4
n
-
7
+
y
4
n
-
6
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
d
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
·
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
·
d
c
d
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
+
c
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
/
c
F
2
n
-
2
+
d
F
2
n
-
3
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
/
c
F
2
n
-
2
+
d
F
2
n
-
3
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
+
c
F
2
n
-
1
+
d
F
2
n
-
2
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
+
d
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
c
F
2
n
+
d
F
2
n
-
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
+
d
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
c
F
2
n
+
1
+
d
F
2
n
c
F
2
n
+
d
F
2
n
-
1
=
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
.Also, we see from this that(53)
y
4
n
-
1
=
y
4
n
-
2
+
y
4
n
-
2
y
4
n
-
5
y
4
n
-
6
+
y
4
n
-
5
=
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
b
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
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0
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·
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·
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·
c
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·
1
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2
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b
+
c
⋯
b
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2
n
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2
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2
n
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4
+
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2
n
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5
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2
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2
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2
n
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2
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2
n
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c
b
c
2
b
+
c
b
+
c
⋯
b
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2
n
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2
n
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4
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2
n
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4
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2
n
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5
+
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2
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+
c
b
+
c
⋯
b
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2
n
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+
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F
2
n
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2
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4
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2
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2
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2
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2
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2
n
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1
=
c
∏
i
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0
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F
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F
2
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+
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+
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F
2
i
+
1
.
Also,(54)y4n=y4n-1+y4n-1y4n-4y4n-5+y4n-4=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·∏i=0n-2bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+a2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5aF2n-1+bF2n-2aF2n-2+bF2n-3·bab2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5+a2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5aF2n-1+bF2n-2aF2n-2+bF2n-3-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+aF2n-1+bF2n-2/aF2n-2+bF2n-31+aF2n-1+bF2n-2/aF2n-2+bF2n-3=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+aF2n-1+bF2n-2aF2n-2+bF2n-3+aF2n-1+bF2n-2=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+11+aF2n-1+bF2n-2aF2n+bF2n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·aF2n+bF2n-1+aF2n-1+bF2n-2aF2n+bF2n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1aF2n+1+bF2naF2n+bF2n-1=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1.
This completes the proof.


### 4.2. Second Case Whenα=A=B=1 and β=-1

Theorem 17.
Let{yn}n=-4∞ be a solution of rational difference equation(55)yn+1=yn-ynyn-3yn-4+yn-3.Then, for n=0,1,2,…, we see that (56)y4n=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1,y4n+1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2,y4n+2=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2,y4n+3=abcdeaFn+bFn+1bFn+1+cFn+2cFn+1+dFn+2dFn+1+eFn+2,where y-4=e,y-3=d,y-2=c,y-1=b,y0=a are arbitrary nonzero real numbers and {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is, (57)y4n-5=abcdeaFn-2+bFn-1bFn-1+cFncFn-1+dFndFn-1+eFn,y4n-4=abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn,y4n-3=abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1,y4n-2=abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1,y4n-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1.
Now, it follows from (55) that(58)y4n=y4n-1-y4n-1y4n-4y4n-5+y4n-4=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn·abcdeaFn-2+bFn-1bFn-1+cFncFn-1+dFndFn-1+eFn+abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+11aFn-1+bFn1aFn-2+bFn-1+1aFn-1+bFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+11aFn-1+bFnaFn-1+bFn+aFn-2+bFn-1aFn-2+bFn-1aFn-1+bFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1aFn-2+bFn-1-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-2+bFn-1aFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn+bFn+1-aFn-2-bFn-1aFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn-1+bFnaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1.Also, we see that (59)y4n+1=y4n-y4ny4n-3y4n-4+y4n-3=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn+abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+11dFn+eFn+11dFn-1+eFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+11dFn+eFn+1dFn+eFn+1+dFn-1+eFndFn-1+eFndFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2dFn-1+eFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdedFn-1+eFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdedFn+1+eFn+2-dFn-1-eFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdedFn+eFn+1aFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2.
Also,(60)y4n+2=y4n+1-y4n+1y4n-2y4n-3+y4n-2=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1+abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+21cFn+dFn+11cFn-1+dFn+1cFn+dFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+21cFn+dFn+1cFn+dFn+1+cFn-1+dFncFn-1+dFncFn+dFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2cFn-1+dFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdecFn-1+dFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdecFn+1+dFn+2-cFn-1-dFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdecFn+dFn+1aFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2,and(61)y4n+3=y4n+2-y4n+2y4n-1y4n-2+y4n-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1+abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+21bFn+cFn+11bFn-1+cFn+1bFn+cFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+21bFn+cFn+1bFn+cFn+1+bFn-1+cFnbFn-1+cFnbFn+cFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2bFn-1+cFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdebFn-1+cFnaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+1+cFn+2-bFn-1-cFnaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+cFn+1aFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+cFn+1aFn+bFn+1bFn+1+cFn+2cFn+1+dFn+2dFn+1+eFn+2.This completes the proof.Example 18.
Figure5 illustrates the solutions of (55) when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 5
Sketch the behavior of the solution of (55).


### 4.3. Third Case Whenα=A=1, β=B=-1

Theorem 19.
Assume that{yn}n=-4∞ is solutions of rational difference equation(62)yn+1=yn-ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (63)y4n=a∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+3-dFi+1cFi+1-dFi-1dFi+3-eFi+1dFi+1-eFi-1,y4n+1=a2d-ed-e∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+3-dFi+1cFi+1-dFi-1dFi+4-eFi+2dFi+2-eFi,y4n+2=a2d-ed-e2c-dc-d∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+4-dFi+2cFi+2-dFidFi+4-eFi+2dFi+2-eFi,y4n+3=a2d-ed-e2c-dc-d2b-cb-c∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+4-cFi+2bFi+2-cFicFi+4-dFi+2cFi+2-dFidFi+4-eFi+2dFi+2-eFi,where y-4=e, y-3=d, y-2=c, y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠y-3≠y-2≠y-1≠y0, {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0, and ∏i=0-1Ai=1.Proof.
The proof of Theorem16 will be omitted.


### 4.4. Fourth Case Whenα=β=A=1 and B=-1

Theorem 20.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(64)yn+1=yn+ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (65)y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b,y12n=a,y12n+1=-aed-e,y12n+2=adec-dd-e,y12n+3=-acdeb-cc-dd-e,y12n+4=abcdea-bb-cc-dd-e,y12n+5=abcea-bb-cc-d,y12n+6=abea-bb-c,y12n+7=aea-b,where y-4=e,y-3=d,y-2=c,y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠y-3≠y-2≠y-1≠y0.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,(66)y12n-12=a,y12n-11=-aed-e,y12n-10=adec-dd-e,y12n-9=-acdeb-cc-dd-e,y12n-8=abcdea-bb-cc-dd-e,y12n-7=abcea-bb-cc-d,y12n-6=abea-bb-c,y12n-5=aea-b,y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b.
Now, it follows from (64) that (67)y12n-4=y12n-5+y12n-5y12n-8y12n-9-y12n-8=aea-b+ae/a-babcde/a-bb-cc-dd-e-acde/b-cc-dd-e-abcde/a-bb-cc-dd-e=aea-b+ae/a-babcde/a-bb-cc-dd-e-a2cde+abcde-abcde/a-bb-cc-dd-e=aea-b-eba-b=ea-ba-b=e,y12n-3=y12n-4+y12n-4y12n-7y12n-8-y12n-7=e+eabce/a-bb-cc-dabcde/a-bb-cc-dd-e-abce/a-bb-cc-d=e+eabce/a-bb-cc-dabcde-abcde+abce2/a-bb-cc-dd-e=e+d-e=d,y12n-2=y12n-3+y12n-3y12n-6y12n-7-y12n-6=d+dabe/a-bb-cabce/a-bb-cc-d-abe/a-bb-c=d+dabe/a-bb-cabce-abce+abcd/a-bb-cc-d=d+dabe/a-bb-cabcd/a-bb-cc-d=d+c-d=c,y12n-1=y12n-2+y12n-2y12n-5y12n-6-y12n-5=c+cae/a-babe/a-bb-c-ae/a-b=c+cae/a-babe-abe+ace/a-bb-c=c+b-c=b.y12n=y12n-1+y12n-1y12n-4y12n-5-y12n-4=b+beae/a-b-e=b+beae-ae+eb/a-b=b+bea-beb=b+a-b=a,y12n+1=y12n+y12ny12n-3y12n-4-y12n-3=a+ade-d=ae-ad+ade-d=aee-d=-aed-e,y12n+2=y12n+1+y12n+1y12n-2y12n-3-y12n-2=-aed-e+-ae/d-ecd-c=-aed-e-aced-ed-c=-aed-e1+cd-c=-aed-ed-c+cd-c=-aedd-ed-c=aedc-dd-e,y12n+3=y12n+2+y12n+2y12n-1y12n-2-y12n-1=aedc-dd-e+aed/c-dd-ebc-b=aedc-dd-e+abedc-dd-ec-b=aedc-dd-e1+bc-b=aedc-dd-ec-b+bc-b=acedc-dd-ec-b=-acedb-cc-dd-e,y12n+4=y12n+3+y12n+3y12ny12n-1-y12n=-acedb-cc-dd-e+-aced/b-cc-dd-eab-a=-acedb-cc-dd-e-a2cedb-cc-dd-eb-a=-acedb-cc-dd-e1+ab-a=-acedb-cc-dd-eb-a+ab-a=-abcedb-cc-dd-eb-a=abceda-bb-cc-dd-e,y12n+5=y12n+4+y12n+4y12n+1y12n-y12n+1=abceda-bb-cc-dd-e+abced/a-bb-cc-dd-e-ae/d-ea--ae/d-e=abceda-bb-cc-dd-e+abced/a-bb-cc-dd-e-ae/d-ead-ae+ae/d-e=abceda-bb-cc-dd-e-abce2a-bb-cc-dd-e=abcea-bb-cc-dd-ed-e=abcea-bb-cc-d,y12n+6=y12n+5+y12n+5y12n+2y12n+1-y12n+2=abcea-bb-cc-d+abce/a-bb-cc-dade/c-dd-e-ae/d-e-ade/c-dd-e=abcea-bb-cc-d+abce/a-bb-cc-dade/c-dd-e-ace+ade-ade/c-dd-e=abcea-bb-cc-d-abeda-bb-cc-d=abea-bb-cc-dc-d=abea-bb-c,y12n+7=y12n+6+y12n+6y12n+3y12n+2-y12n+3=abea-bb-c+abe/a-bb-c-acde/b-cc-dd-eade/c-dd-e--acde/b-cc-dd-e=abea-bb-c+abe/a-bb-c-acde/b-cc-dd-eabde-acde+acde/b-cc-dd-e=abea-bb-c-acea-bb-c=aea-bb-cb-c=aea-b,
This completes the proof.Theorem 21.
Every solution{yn}n=-4∞ of rational difference equation (64) is periodic with period 12.Proof.
The proof follows from the previous theorem and will be omitted.Example 22.
Figure6 shows that the solution of (64) has a periodic solution with period 12 when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3,y-1=0.5, and y0=0.1.Figure 6
Plot the periodicity of the solution of (64).


### 4.5. Fifth Case Whenβ=A=B=1 and α=-1

Theorem 23.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(68)yn+1=-yn+ynyn-3yn-4+yn-3.Then every solution of (68) is periodic with period 12. Moreover {yn}n=-4∞ takes the form (69)y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b,y12n=a,y12n+1=-aed+e,y12n+2=adec+dd+e,y12n+3=-acdeb+cc+dd+ey12n+4=abcdea+bb+cc+dd+e,y12n+5=abcea+bb+cc+d,y12n+6=abea+bb+c,y12n+7=aea+b,where y-4=e, y-3=d, y-2=c, y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠-y-3, y-3≠-y-2, y-2≠-y-1, and y-1≠-y0.Proof.
The proof is left to the reader.Example 24.
Figure7 shows that the solution of (68) has a periodic solution with period 12 when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 7
Plot the periodicity of the solution of (68).


### 4.6. Sixth Case Whenβ=A=1 and α=B=-1

Theorem 25.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(70)yn+1=-yn+ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (71)y4n-3=-d∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+1-cF2ibF2i-cF2i-1cF2i+1-dF2icF2i-dF2i-1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n-2=c∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+1-cF2ibF2i-cF2i-1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n-1=-b∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+3-cF2i+2bF2i+2-cF2i+1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n=a∏i=0n-1aF2i+3-bF2i+2aF2i+2-bF2i+1bF2i+3-cF2i+2bF2i+2-cF2i+1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,where y-4=e, y-3=d, y-2=c, y-1=b, y0=a, y-4≠y-3≠y-2≠y-1≠y0, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.


### 4.7. Seventh Case Whenβ=B=1 and α=A=-1

Theorem 26.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(72)yn+1=-yn+ynyn-3yn-3-yn-4,and then for n=0,1,2,…, we see that (73)y4n=abcdeaFn-bFn+1bFn-cFn+1cFn-dFn+1dFn-eFn+1,y4n+1=abcdeaFn-bFn+1bFn-cFn+1cFn-dFn+1dFn+1-eFn+2,y4n+2=abcdeaFn-bFn+1bFn-cFn+1cFn+1-dFn+2dFn+1-eFn+2,y4n+3=abcdeaFn-bFn+1bFn+1-cFn+2cFn+1-dFn+2dFn+1-eFn+2,where y-4=e,y-3=d,y-2=c,y-1=b, and y0=a are arbitrary nonzero real numbers y-4≠y-3≠y-2≠y-1≠y0 and {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0.


### 4.8. Eighth Case WhenA=B=1 and α=β=-1

Theorem 27.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(74)yn+1=-yn-ynyn-3yn-4+yn-3,and then, for n=0,1,2,…, we see that (75)y4n-3=-d∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-2=c∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-1=-b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,where y-4=e,y-3=d,y-2=c,y-1=b,y0=a, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.


## 4.1. First Case Whenα=β=A=B=1

Theorem 16.
Suppose that{yn}n=-4∞ is a solution of rational difference equation(48)yn+1=yn+ynyn-3yn-4+yn-3.Then, for n=0,1,2,…, we see that (49)y4n-3=d∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-2=c∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,where y-4=e,y-3=d, y-2=c, y-1=b, y0=a, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,(50)y4n-8=a∏i=0n-3aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-7=d∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-6=c∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-5=b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-4=a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1.
From (48), we see that (51)
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+
3
d
3
c
+
d
⋯
a
F
2
n
-
3
+
b
F
2
n
-
4
a
F
2
n
-
4
+
b
F
2
n
-
5
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
a
b
b
c
c
d
2
a
+
b
a
+
b
2
b
+
c
b
+
c
2
c
+
d
c
+
d
5
a
+
3
b
3
a
+
b
5
b
+
3
c
3
b
+
c
5
c
+
3
d
3
c
+
d
⋯
a
F
2
n
-
3
+
b
F
2
n
-
4
a
F
2
n
-
4
+
b
F
2
n
-
5
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
d
a
b
b
c
c
d
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
a
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
a
b
b
c
c
d
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
a
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
a
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
a
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
∏
i
=
0
n
-
3
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
∏
i
=
0
n
-
2
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
/
d
F
2
n
-
2
+
e
F
2
n
-
3
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
/
d
F
2
n
-
2
+
e
F
2
n
-
3
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
-
2
+
e
F
2
n
-
3
+
d
F
2
n
-
1
+
e
F
2
n
-
2
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
+
e
F
2
n
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
F
2
n
+
e
F
2
n
-
1
+
d
F
2
n
-
1
+
e
F
2
n
-
2
d
F
2
n
+
e
F
2
n
-
1
=
a
∏
i
=
0
n
-
2
a
F
2
i
+
3
+
b
F
2
i
+
2
a
F
2
i
+
2
+
b
F
2
i
+
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
F
2
n
+
1
+
e
F
2
n
d
F
2
n
+
e
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
.
(52)
y
4
n
-
2
=
y
4
n
-
3
+
y
4
n
-
3
y
4
n
-
6
y
4
n
-
7
+
y
4
n
-
6
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
d
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
·
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
∏
i
=
0
n
-
2
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
+
c
∏
i
=
0
n
-
2
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
·
d
c
d
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
+
c
2
c
+
d
c
+
d
⋯
c
F
2
n
-
3
+
d
F
2
n
-
4
c
F
2
n
-
4
+
d
F
2
n
-
5
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
/
c
F
2
n
-
2
+
d
F
2
n
-
3
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
/
c
F
2
n
-
2
+
d
F
2
n
-
3
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
-
2
+
d
F
2
n
-
3
+
c
F
2
n
-
1
+
d
F
2
n
-
2
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
+
d
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
c
F
2
n
+
d
F
2
n
-
1
+
c
F
2
n
-
1
+
d
F
2
n
-
2
c
F
2
n
+
d
F
2
n
-
1
=
d
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
1
+
d
F
2
i
c
F
2
i
+
d
F
2
i
-
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
c
F
2
n
+
1
+
d
F
2
n
c
F
2
n
+
d
F
2
n
-
1
=
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
.Also, we see from this that(53)
y
4
n
-
1
=
y
4
n
-
2
+
y
4
n
-
2
y
4
n
-
5
y
4
n
-
6
+
y
4
n
-
5
=
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
b
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
c
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
b
∏
i
=
0
n
-
2
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
3
+
c
F
2
i
+
2
b
F
2
i
+
2
+
c
F
2
i
+
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
-
1
=
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
+
c
∏
i
=
0
n
-
1
a
F
2
i
+
1
+
b
F
2
i
a
F
2
i
+
b
F
2
i
-
1
b
F
2
i
+
1
+
c
F
2
i
b
F
2
i
+
c
F
2
i
-
1
c
F
2
i
+
3
+
d
F
2
i
+
2
c
F
2
i
+
2
+
d
F
2
i
+
1
d
F
2
i
+
3
+
e
F
2
i
+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
·
b
∏
i
=
0
n
-
2
a
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2
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+
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+
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2
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+
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2
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·
∏
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=
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·
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·
c
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1
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+
2
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F
2
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+
1
·
1
+
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2
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+
c
b
+
c
⋯
b
F
2
n
-
3
+
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2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
b
F
2
n
-
1
+
c
F
2
n
-
2
b
F
2
n
-
2
+
c
F
2
n
-
3
c
b
c
2
b
+
c
b
+
c
⋯
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
c
F
2
n
-
5
+
b
2
b
+
c
b
+
c
⋯
b
F
2
n
-
3
+
c
F
2
n
-
4
b
F
2
n
-
4
+
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F
2
n
-
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F
2
n
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1
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2
n
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2
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2
n
-
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+
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F
2
n
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3
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1
=
c
∏
i
=
0
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2
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+
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/
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2
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2
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1
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n
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=
c
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0
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2
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2
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F
2
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+
2
d
F
2
i
+
2
+
e
F
2
i
+
1
.
Also,(54)y4n=y4n-1+y4n-1y4n-4y4n-5+y4n-4=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·∏i=0n-2bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1+b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1·b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1b∏i=0n-2aF2i+1+bF2iaF2i+bF2i-1+a∏i=0n-2aF2i+3+bF2i+2aF2i+2+bF2i+1-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+a2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5aF2n-1+bF2n-2aF2n-2+bF2n-3·bab2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5+a2a+ba+b⋯aF2n-3+bF2n-4aF2n-4+bF2n-5aF2n-1+bF2n-2aF2n-2+bF2n-3-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+aF2n-1+bF2n-2/aF2n-2+bF2n-31+aF2n-1+bF2n-2/aF2n-2+bF2n-3=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·1+aF2n-1+bF2n-2aF2n-2+bF2n-3+aF2n-1+bF2n-2=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+11+aF2n-1+bF2n-2aF2n+bF2n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1·aF2n+bF2n-1+aF2n-1+bF2n-2aF2n+bF2n-1=b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1aF2n+1+bF2naF2n+bF2n-1=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1.
This completes the proof.


## 4.2. Second Case Whenα=A=B=1 and β=-1

Theorem 17.
Let{yn}n=-4∞ be a solution of rational difference equation(55)yn+1=yn-ynyn-3yn-4+yn-3.Then, for n=0,1,2,…, we see that (56)y4n=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1,y4n+1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2,y4n+2=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2,y4n+3=abcdeaFn+bFn+1bFn+1+cFn+2cFn+1+dFn+2dFn+1+eFn+2,where y-4=e,y-3=d,y-2=c,y-1=b,y0=a are arbitrary nonzero real numbers and {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is, (57)y4n-5=abcdeaFn-2+bFn-1bFn-1+cFncFn-1+dFndFn-1+eFn,y4n-4=abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn,y4n-3=abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1,y4n-2=abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1,y4n-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1.
Now, it follows from (55) that(58)y4n=y4n-1-y4n-1y4n-4y4n-5+y4n-4=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn·abcdeaFn-2+bFn-1bFn-1+cFncFn-1+dFndFn-1+eFn+abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+11aFn-1+bFn1aFn-2+bFn-1+1aFn-1+bFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+11aFn-1+bFnaFn-1+bFn+aFn-2+bFn-1aFn-2+bFn-1aFn-1+bFn-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1aFn-2+bFn-1-1=abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn-2+bFn-1aFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn+bFn+1-aFn-2-bFn-1aFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn-1+bFnaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1aFn+bFn+1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1.Also, we see that (59)y4n+1=y4n-y4ny4n-3y4n-4+y4n-3=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn-1+eFn+abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+11dFn+eFn+11dFn-1+eFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+11dFn+eFn+1dFn+eFn+1+dFn-1+eFndFn-1+eFndFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2dFn-1+eFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1-abcdedFn-1+eFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdedFn+1+eFn+2-dFn-1-eFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdedFn+eFn+1aFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+eFn+1dFn+1+eFn+2=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2.
Also,(60)y4n+2=y4n+1-y4n+1y4n-2y4n-3+y4n-2=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn-1+dFndFn+eFn+1+abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+21cFn+dFn+11cFn-1+dFn+1cFn+dFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+21cFn+dFn+1cFn+dFn+1+cFn-1+dFncFn-1+dFncFn+dFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2cFn-1+dFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2-abcdecFn-1+dFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdecFn+1+dFn+2-cFn-1-dFnaFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdecFn+dFn+1aFn+bFn+1bFn+cFn+1cFn+dFn+1dFn+1+eFn+2cFn+1+dFn+2=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2,and(61)y4n+3=y4n+2-y4n+2y4n-1y4n-2+y4n-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1·abcdeaFn-1+bFnbFn-1+cFncFn+dFn+1dFn+eFn+1+abcdeaFn-1+bFnbFn+cFn+1cFn+dFn+1dFn+eFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+21bFn+cFn+11bFn-1+cFn+1bFn+cFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+21bFn+cFn+1bFn+cFn+1+bFn-1+cFnbFn-1+cFnbFn+cFn+1-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2bFn-1+cFn-1=abcdeaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2-abcdebFn-1+cFnaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+1+cFn+2-bFn-1-cFnaFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+cFn+1aFn+bFn+1bFn+cFn+1cFn+1+dFn+2dFn+1+eFn+2bFn+1+cFn+2=abcdebFn+cFn+1aFn+bFn+1bFn+1+cFn+2cFn+1+dFn+2dFn+1+eFn+2.This completes the proof.Example 18.
Figure5 illustrates the solutions of (55) when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 5
Sketch the behavior of the solution of (55).


## 4.3. Third Case Whenα=A=1, β=B=-1

Theorem 19.
Assume that{yn}n=-4∞ is solutions of rational difference equation(62)yn+1=yn-ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (63)y4n=a∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+3-dFi+1cFi+1-dFi-1dFi+3-eFi+1dFi+1-eFi-1,y4n+1=a2d-ed-e∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+3-dFi+1cFi+1-dFi-1dFi+4-eFi+2dFi+2-eFi,y4n+2=a2d-ed-e2c-dc-d∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+3-cFi+1bFi+1-cFi-1cFi+4-dFi+2cFi+2-dFidFi+4-eFi+2dFi+2-eFi,y4n+3=a2d-ed-e2c-dc-d2b-cb-c∏i=0n-1aFi+3-bFi+1aFi+1-bFi-1bFi+4-cFi+2bFi+2-cFicFi+4-dFi+2cFi+2-dFidFi+4-eFi+2dFi+2-eFi,where y-4=e, y-3=d, y-2=c, y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠y-3≠y-2≠y-1≠y0, {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0, and ∏i=0-1Ai=1.Proof.
The proof of Theorem16 will be omitted.


## 4.4. Fourth Case Whenα=β=A=1 and B=-1

Theorem 20.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(64)yn+1=yn+ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (65)y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b,y12n=a,y12n+1=-aed-e,y12n+2=adec-dd-e,y12n+3=-acdeb-cc-dd-e,y12n+4=abcdea-bb-cc-dd-e,y12n+5=abcea-bb-cc-d,y12n+6=abea-bb-c,y12n+7=aea-b,where y-4=e,y-3=d,y-2=c,y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠y-3≠y-2≠y-1≠y0.Proof.
Forn=0 the result holds. Now, suppose that n>0 and that our assumption holds for n-1. That is,(66)y12n-12=a,y12n-11=-aed-e,y12n-10=adec-dd-e,y12n-9=-acdeb-cc-dd-e,y12n-8=abcdea-bb-cc-dd-e,y12n-7=abcea-bb-cc-d,y12n-6=abea-bb-c,y12n-5=aea-b,y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b.
Now, it follows from (64) that (67)y12n-4=y12n-5+y12n-5y12n-8y12n-9-y12n-8=aea-b+ae/a-babcde/a-bb-cc-dd-e-acde/b-cc-dd-e-abcde/a-bb-cc-dd-e=aea-b+ae/a-babcde/a-bb-cc-dd-e-a2cde+abcde-abcde/a-bb-cc-dd-e=aea-b-eba-b=ea-ba-b=e,y12n-3=y12n-4+y12n-4y12n-7y12n-8-y12n-7=e+eabce/a-bb-cc-dabcde/a-bb-cc-dd-e-abce/a-bb-cc-d=e+eabce/a-bb-cc-dabcde-abcde+abce2/a-bb-cc-dd-e=e+d-e=d,y12n-2=y12n-3+y12n-3y12n-6y12n-7-y12n-6=d+dabe/a-bb-cabce/a-bb-cc-d-abe/a-bb-c=d+dabe/a-bb-cabce-abce+abcd/a-bb-cc-d=d+dabe/a-bb-cabcd/a-bb-cc-d=d+c-d=c,y12n-1=y12n-2+y12n-2y12n-5y12n-6-y12n-5=c+cae/a-babe/a-bb-c-ae/a-b=c+cae/a-babe-abe+ace/a-bb-c=c+b-c=b.y12n=y12n-1+y12n-1y12n-4y12n-5-y12n-4=b+beae/a-b-e=b+beae-ae+eb/a-b=b+bea-beb=b+a-b=a,y12n+1=y12n+y12ny12n-3y12n-4-y12n-3=a+ade-d=ae-ad+ade-d=aee-d=-aed-e,y12n+2=y12n+1+y12n+1y12n-2y12n-3-y12n-2=-aed-e+-ae/d-ecd-c=-aed-e-aced-ed-c=-aed-e1+cd-c=-aed-ed-c+cd-c=-aedd-ed-c=aedc-dd-e,y12n+3=y12n+2+y12n+2y12n-1y12n-2-y12n-1=aedc-dd-e+aed/c-dd-ebc-b=aedc-dd-e+abedc-dd-ec-b=aedc-dd-e1+bc-b=aedc-dd-ec-b+bc-b=acedc-dd-ec-b=-acedb-cc-dd-e,y12n+4=y12n+3+y12n+3y12ny12n-1-y12n=-acedb-cc-dd-e+-aced/b-cc-dd-eab-a=-acedb-cc-dd-e-a2cedb-cc-dd-eb-a=-acedb-cc-dd-e1+ab-a=-acedb-cc-dd-eb-a+ab-a=-abcedb-cc-dd-eb-a=abceda-bb-cc-dd-e,y12n+5=y12n+4+y12n+4y12n+1y12n-y12n+1=abceda-bb-cc-dd-e+abced/a-bb-cc-dd-e-ae/d-ea--ae/d-e=abceda-bb-cc-dd-e+abced/a-bb-cc-dd-e-ae/d-ead-ae+ae/d-e=abceda-bb-cc-dd-e-abce2a-bb-cc-dd-e=abcea-bb-cc-dd-ed-e=abcea-bb-cc-d,y12n+6=y12n+5+y12n+5y12n+2y12n+1-y12n+2=abcea-bb-cc-d+abce/a-bb-cc-dade/c-dd-e-ae/d-e-ade/c-dd-e=abcea-bb-cc-d+abce/a-bb-cc-dade/c-dd-e-ace+ade-ade/c-dd-e=abcea-bb-cc-d-abeda-bb-cc-d=abea-bb-cc-dc-d=abea-bb-c,y12n+7=y12n+6+y12n+6y12n+3y12n+2-y12n+3=abea-bb-c+abe/a-bb-c-acde/b-cc-dd-eade/c-dd-e--acde/b-cc-dd-e=abea-bb-c+abe/a-bb-c-acde/b-cc-dd-eabde-acde+acde/b-cc-dd-e=abea-bb-c-acea-bb-c=aea-bb-cb-c=aea-b,
This completes the proof.Theorem 21.
Every solution{yn}n=-4∞ of rational difference equation (64) is periodic with period 12.Proof.
The proof follows from the previous theorem and will be omitted.Example 22.
Figure6 shows that the solution of (64) has a periodic solution with period 12 when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3,y-1=0.5, and y0=0.1.Figure 6
Plot the periodicity of the solution of (64).


## 4.5. Fifth Case Whenβ=A=B=1 and α=-1

Theorem 23.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(68)yn+1=-yn+ynyn-3yn-4+yn-3.Then every solution of (68) is periodic with period 12. Moreover {yn}n=-4∞ takes the form (69)y12n-4=e,y12n-3=d,y12n-2=c,y12n-1=b,y12n=a,y12n+1=-aed+e,y12n+2=adec+dd+e,y12n+3=-acdeb+cc+dd+ey12n+4=abcdea+bb+cc+dd+e,y12n+5=abcea+bb+cc+d,y12n+6=abea+bb+c,y12n+7=aea+b,where y-4=e, y-3=d, y-2=c, y-1=b, and y0=a are arbitrary nonzero real numbers with y-4≠-y-3, y-3≠-y-2, y-2≠-y-1, and y-1≠-y0.Proof.
The proof is left to the reader.Example 24.
Figure7 shows that the solution of (68) has a periodic solution with period 12 when the initial conditions y-4=0.57, y-3=1.1, y-2=0.3, y-1=0.5, and y0=0.1.Figure 7
Plot the periodicity of the solution of (68).


## 4.6. Sixth Case Whenβ=A=1 and α=B=-1

Theorem 25.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(70)yn+1=-yn+ynyn-3yn-4-yn-3,and then for n=0,1,2,…, we see that (71)y4n-3=-d∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+1-cF2ibF2i-cF2i-1cF2i+1-dF2icF2i-dF2i-1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n-2=c∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+1-cF2ibF2i-cF2i-1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n-1=-b∏i=0n-1aF2i+1-bF2iaF2i-bF2i-1bF2i+3-cF2i+2bF2i+2-cF2i+1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,y4n=a∏i=0n-1aF2i+3-bF2i+2aF2i+2-bF2i+1bF2i+3-cF2i+2bF2i+2-cF2i+1cF2i+3-dF2i+2cF2i+2-dF2i+1dF2i+3-eF2i+2dF2i+2-eF2i+1,where y-4=e, y-3=d, y-2=c, y-1=b, y0=a, y-4≠y-3≠y-2≠y-1≠y0, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.


## 4.7. Seventh Case Whenβ=B=1 and α=A=-1

Theorem 26.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(72)yn+1=-yn+ynyn-3yn-3-yn-4,and then for n=0,1,2,…, we see that (73)y4n=abcdeaFn-bFn+1bFn-cFn+1cFn-dFn+1dFn-eFn+1,y4n+1=abcdeaFn-bFn+1bFn-cFn+1cFn-dFn+1dFn+1-eFn+2,y4n+2=abcdeaFn-bFn+1bFn-cFn+1cFn+1-dFn+2dFn+1-eFn+2,y4n+3=abcdeaFn-bFn+1bFn+1-cFn+2cFn+1-dFn+2dFn+1-eFn+2,where y-4=e,y-3=d,y-2=c,y-1=b, and y0=a are arbitrary nonzero real numbers y-4≠y-3≠y-2≠y-1≠y0 and {Fi}i=0∞=1,1,2,3,5,8,13,…, F0=0.


## 4.8. Eighth Case WhenA=B=1 and α=β=-1

Theorem 27.
Suppose that{yn}n=-4∞ is solutions of rational difference equation(74)yn+1=-yn-ynyn-3yn-4+yn-3,and then, for n=0,1,2,…, we see that (75)y4n-3=-d∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+1+dF2icF2i+dF2i-1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-2=c∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+1+cF2ibF2i+cF2i-1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n-1=-b∏i=0n-1aF2i+1+bF2iaF2i+bF2i-1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,y4n=a∏i=0n-1aF2i+3+bF2i+2aF2i+2+bF2i+1bF2i+3+cF2i+2bF2i+2+cF2i+1cF2i+3+dF2i+2cF2i+2+dF2i+1dF2i+3+eF2i+2dF2i+2+eF2i+1,where y-4=e,y-3=d,y-2=c,y-1=b,y0=a, {Fi}i=1∞=1,1,2,3,5,8,13,…, F-1=1,F0=0, and ∏i=0-1Ai=1.



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*Source: 9129354-2018-07-19.xml*