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Periodicity of Generalized Fibonacci-like Sequences

Alexander V. Evako

“Dianet”, Laboratory of Digital Technologies, Moscow, Russia

Abstract: Generalized Fibonacci-like sequences appear in finite difference approximations of the Partial Differential Equations based upon replacing partial differential equations by finite difference equations. This paper studies properties of the generalized Fibonacci-like sequence Fn+2=A+BFn+1+CFn. It is shown that this sequence is periodic with period T>2, if C= -1, |B|<2.

Keywords: generalized fibonacci-like sequence, periodicity, discrete dynamical system, PDE, graph, network, digital space

Citation: Alexander V. Evako, Periodicity of Generalized Fibonacci-like Sequences. Applied Mathematics and Physics, 5 (1). 11-18. January 2017. doi:10.12691/amp-5-1-2

1. Introduction

Fibonacci sequences have attracted attention of scientists from different fields. Fibonacci numbers are widely used in many areas from applied mathematics to physics. The Fibonacci sequence, Lucas sequence, Pell sequence and periodic generalized Fibonacci sequences modulo m have been intensively studied for many years. Over the last decades, numerous useful theoretical results have been obtained in this field (see, e.g., [1,10,15,17,21,22]).

In this paper, we introduce and study a new generalization of Fibonacci-like sequences; we are interested in the periodicity of the Fibonacci-like sequence.

The essential fact is that finite difference PDE equations on discrete domains contain generalized Fibonacci-like sequences as basic elements of these equations, and knowledge of features of these sequences plays an important role in analyzing properties computational solutions of PDE such as the period and the frequency of oscillations.

Finite difference approximations of the PDE are based upon replacing partial differential equations by finite difference equations [13,18]. In the past decades, computational solutions of PDE on various domains including non-orientable surfaces such as a Moebius strip and a Klein bottle have been studied by scientists from different fields. The study was derived from obvious practical and science background. In physics, a considerable interest has emerged in studying lattice models on non-orientable surfaces as new challenging unsolved lattice-statistical problems and as a realization and testing of predictions of the conformal field theory (see, e.g., [12]). Many important technical and physical properties of non-orientable structural elements can be described by solutions of partial differential equations (PDE), where a non-orientable space serves as a domain. A problem exists in description of electronic and nuclear motions in nano-technology structures and biological networks. Modeling blood flow through a capillary network or road traffic requires a system of differential equations on a graph or network . Since analytic solutions of PDE can be obtained only in simple geometric regions, for practical problems, it is more reasonable to use computational or numerical solutions. It can be done by implementing as domains graphs, networks and digital spaces, which are discreet counterparts of continuous spaces, and by transferring PDE from a continuous area into discrete one. A review of works devoted to partial differential equations on networks and graphs can be found in [4,14,16].

The material to be presented in Section 2 begins with the investigation of properties of a periodic continuous function , where . It is shown that if is periodic with period then .

In section 3, we study properties of the sequence, which is defined by recurrence relation with , where A, B C, a and b are real numbers. We show that if then is periodic with the period .

Section 4 presents numerical examples of periodic Fibonacci-like sequences and gives examples of application of Fibonacci-like sequences to the numerical solutions of the wave equation on the graph containing two vertices and on the graph containing sixteen vertices and presenting a non-orientable Klein bottle, which is an important object of study in physics. Using properties of Fibonacci-like sequences, we show that the period of oscillations on the two-point graph is T=13.9. Oscillations at different points of the Klein bottle are also periodic.

2. Periodicity of Fibonacci-like Continuous Functions

In a numerical solution of the one-dimensional wave equation by the finite difference method, the spatial domain and the temporal domain are replaced by a set of mesh points, the second-order derivatives can be replaced by central differences [4,13,14,16]. The solution of the wave equation is replaced by the mesh function, which approximates the exact solution at the mesh points. The obtained discretization scheme has the form , where .

Note that the solution is periodic in time. Using this formula, consider properties of periodic continuous function f(x) on domain R provided that .

Theorem 1

Let f(x) be a continuous function on domain R, and let

[picture] | (1)

for any . If f(x) is a periodic function with period then , , .

Proof.

Let f(x) is periodic with period . Consider the Fourier series representing f(x).

[picture] | (2)

Using (2), it is easy to see that

where

[picture] | (3)

[picture] | (4)

[picture] | (5)

Since

then for any n, coefficients c₀, cn and dn are equal to zero.

[picture] | (6)

Let n=1.

As f(x) is periodic with basic period then and/or , which means that

[picture] | (7)

Suppose that , i.e., . It follows from (7) that Therefore, . Hence,

[picture] | (8)

Let n>1.

For the same reason as above, if then , if then and/or . Note that if

Finally, find a₀ from equation (3). Notice that .

Thus, if and then f(x) is periodic with basic period and .

[picture] | (9)

where if . This completes the proof.

Theorem 2

Let f(x) be a continuous periodic function on domain R with period, and for any ,

Then if .

if .

Proof

Let k is even, i.e., k=2n and . Then

This means that in (9),

Let k is odd, i.e., k=2n+1 and . Then and Since , then by (7), . Therefore,

[picture] | (10)

It follows from (10) that . This completes the proof.

We now outline a correspondence between the number B and the period T of function , where . Let’s investigate this question graphically. The graph is shown in Figure 1. Obviously, .

Figure 1. The period profile T of function f(x) and sequence Fn, |B|<2

3. Periodic generalized Fibonacci-like Sequences

Periodic generalized Fibonacci sequences modulo m have been studied in a number of papers (see e.g. [9,11,19,21]). In this section, we consider the periodic generalized Fibonacci sequence

[picture] | (11)

In sequence (11), and are real numbers. In particular, and can be integer. The results of this section are directly deduced from the previous section. The following theorems are a direct consequence of theorems 1 and 2.

Theorem 3

Let , be a sequence of real numbers, If then is a periodic sequence with period,

[picture] | (12)

Proof.

Consider the function where . Then for .

According to theorem 1, if , then f(x) is periodic with period , . . Therefore, sequence is also periodic with period . This completes the proof.

Consider a special case when the period T=1, 2.

Theorem 4

Let , be a sequence of real numbers, .

If then is a periodic sequence with period , .

If then is a periodic sequence with period ,

Let f(x) be a continuous periodic function on domain R, and .

If the period of f(x) then by theorem 2. Therefore, .

If the period of f(x) then by theorem 2. Therefore, . This completes the proof.

4. Numerical Examples

In this Section, we consider examples of the periodic generalized Fibonacci sequence and its applications to numerical solution of the wave equation on graphs.

4.1. Numerical Examples of Periodic Fibonacci-like Sequences

for .

Example 1

First, consider the case, when coefficient B is close to 2. Let . Then . In Figure 2, the sequence profile Fn, , is plotted for n=1,…50. The plot illustrates the periodicity of Fn with period , as it follows from formulas (12) , . For the first n=0,1,…13, the terms of Fn are given in Table 1.

Table 1.

Figure 2. Fn profile at n=0, 1,… 50. .

Example 2

In this case, take the following constants: . Then . In Figure 3, we have plotted the sequence profile Fn, , at n=1,…20. The plot shows that Fn is periodic with period T=6. According to (12), , . For the first n=0,1,…13, the terms of Fn are shown in Table 2.

Table 2

Figure 3. Fn profile at n=0, 1,… 20. .

Example 3

Consider the case, when . Then . In Figure 4, the profile Fn, , is plotted for n=1,…20. Fn is periodic with period T=4, as it follows from (12). For the first n=0,1,…13, the terms of Fn are given Table 3.

Table 3

Figure 4. Fn profile at n=0, 1,… 20. .

Example 4

Let . Then . The profile of Fn for n=1,…20 is shown in Figure 5. Fn, , is periodic with period , as it follows from formulas , .

For the first n=0,1,…13, the terms of Fn are given in Table 4.

Table 4

Figure 5. Fn profile at n=0, 1,… 20. .

Example 5

Choose . Then . In Figure 6, we have plotted the sequence profile Fn, , at n=1,…20. The plot shows that Fn is periodic with period T=2.51. According to (12), For the first n=0,1,…13, the terms of sequence Fn are given in Table 5.

Table 5.

Figure 6. The profile of Fn at n=0, 1,… 20. .

4.2. Numerical Examples of Solutions of the Wave Equation Containing Periodic Fibonacci-Like Sequences on Graphs

As it was mentioned above, finite difference approximations of the PDE are based upon replacing partial differential equations by finite difference equations using Taylor approximations [18]. Consider a hyperbolic PDE with two spatial independent variables.

where Using a standard two-dimensional spatial orthogonal grid H with points , the forward difference formula for the derivative with respect to t, and the central difference formula for the second derivatives with respect to x and y, we obtain the following equivalent finite deference equation

[picture] | (13)

where . By construction, grid H is a graph, in which point is adjacent to points and The nearest neighborhood (ball) of point is the subgraph containing and points and . Using these notations, equation (13) can be written as the following.

[picture] | (14)

where .

The set (14) can be applied to a general graph G(V, W) with the set of points V=(v₁, v₂,...vs) and the set of edges W = ((vрvq),....), where is the ball of point , p=1,...s. Set (14) is similar to the set of differential equations on a graph investigated by A. I. Volpert in [20].

The wave equation defined below is a special case of hyperbolic equation (14) if the following condition holds

[picture] | (15)

The solution of this equation should be analogous the solution of the wave equation in the continuous case. It should be associated with oscillations of the function at points of G and the wave propagation.

There is considerable amount of literature devoted to the study of different approaches to digital lines, surfaces and spaces used by researchers. Traditionally, a digital image has a graph structure (see [2,3,5]). A digital space G is a simple undirected graph G=(V,W) where V=(v₁,v₂,...vn,…) is a finite or countable set of points, and W = ((vрvq),....) is a set of edges. The induced subgraph containing point v and all points adjacent to v is called the ball of point v in G. Graphs that are digital counterparts of continuous n-dimensional manifolds were studied in [5,6,7,8]. Consider the numerical solutions of the initial value problem for the wave equation on a graph . We will see that periodic Fibonacci-like sequences are basic elements of this equation.

Example 6. Numerical solution of the initial value problem on a connected graph G with two points.

Consider a connected graph consisting of two adjacent points depicted in Figure 7(a). According to (15)

[picture] | (16)

Let and . Then for any n>1. For set (16) we have two sequences

[picture] | (17)

According to theorem 3, sequence (17) is the generalized Fibonacci-like sequence with . Since , and , then . Therefore, sequence (17) is periodic with period , as it follows from (12). For the same reason, is periodic with the same period T. Define coefficients in equation (16) in the following way; . Initial values are given as , . Therefore, , .

The period of oscillations is . The solution of the initial value problem at points 1 and 2 for t=1,…50 is displayed in Figure 7(b), which illustrates the time behavior of the values of function f. Figure 7(b) presents oscillations at points 1 and 2.

Example 7. Numerical solution of the initial value problem on the digital Klein bottle.

A Klein bottle is an object of investigation in many fields. In physics [12], a Klein bottle attracted attention in studying lattice models on non-orientable surfaces A digital 2D Klein bottle K depicted in Figure 8(a) consists of sixteen points. The nearest neighborhood O(vk) of every point vk is a digital 1-sphere containing six points. Topological properties of K are similar to topological properties of its continuous counterpart. For example, the Euler characteristic and the homology groups of a continuous and a digital Klein bottle are the same ([7] and [8]). Consider the solution of equation (15) on K. One can check directly that all sequences in set (15) are periodic generalized Fibonacci-like sequences. Define coefficients in the following way. If points vp and vk are adjacent then =0.01and ckk=0.94 for k=1,…16. Initial values are given as =3, =3. In Figure 8(b), numerical solutions at points 1 and 3 of the of the initial value problem are plotted at time t=1,…100. Two lines show oscillations with period T>2.

Figure 7. (a) G is a graph with two adjacent points. (b) Numerical solution of the initial value problem on graph G with two points v₁ and v₂. Initial values are f₁⁰=f₁¹=3, f₂⁰=f₂¹=0. The solutions on G are shown at points 1 and 2, t=0, 1,… 50,. The solution profiles are oscillations with period T=13.9

Figure. 8. (a) K is a digital Klein bottle. (b) The solution profiles of the initial value problem on the Klein bottle K at points 1 and 3, t=0, 1,… 100, f₆⁰=f₆¹ =3, fk⁰=fk¹ =0, k≠6. Two lines are oscillations with period T>2

5. Conclusion

In connection with computational methods for approximating the solutions of partial differential equations, the generalized Fibonacci-like sequence with , where A, B, C, a and b are real numbers is investigated. It is shown that if then is periodic with the period . If period , then

If period and then , . Numerical examples of periodic Fibonacci-like sequences, and periodic numerical solutions of the wave equation on graphs are presented.

Competing Interests

The author declares that he has no competing interests.

Acknowledgements

The author has got no financial or any other support from any person, fund, organization, etc.

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