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Journal of Applied Mathematics
Volume 2014, Article ID 491591, 8 pages
http://dx.doi.org/10.1155/2014/491591
Research Article

On Types of Distance Fibonacci Numbers Generated by Number Decompositions

Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszow, Poland

Received 2 June 2014; Revised 21 August 2014; Accepted 23 August 2014; Published 23 October 2014

Academic Editor: Ali R. Ashrafi

Copyright © 2014 Anetta Szynal-Liana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce new types of distance Fibonacci numbers which are closely related with number decompositions. Using special decompositions of the number we give a sequence of identities for them. Moreover, we give matrix generators for distance Fibonacci numbers and their direct formulas.

1. Introduction

The th Fibonacci numbers are defined by recurrence relation , with the initial conditions . There are many generalizations of the Fibonacci numbers with respect to one or more parameters; see for example [13]. In [1] the distance Fibonacci numbers were introduced and studied. We recall this definition.

Let , be integers. The distance Fibonacci numbers of the first kind are defined recursively in the following way: and for .

We will call the numbers the distance Fibonacci numbers of the first kind. The number is closely related to the special quasi -decomposition of the number ; see [1].

In this paper we define other three types of distance Fibonacci numbers which are also related to the special number decomposition. Moreover we shall show relations between all three types of distance Fibonacci numbers. Next we study their matrix generators and direct formulas.

2. Distance Fibonacci Numbers and

In this section we introduce two kinds of distance Fibonacci numbers. Some relations between numbers for will be studied.

Let , be integers. We define the th distance Fibonacci numbers of the second kind by the th order linear recurrence relation of the form with the initial conditionsfor ,,,,, for .

If , , then gives the Fibonacci numbers .

Let , be integers. We define the th distance Fibonacci numbers of the third kind by the th order linear recurrence relation of the form with the initial conditions for ,,for   ,for   .

If , then gives the classical Fibonacci numbers.

Now we give an interpretation of the numbers , , and with respect to special decompositions of the number .

By a decomposition of a number , , we mean an ordered number partition of it. For example for we have the following four decompositions: , , , and 3. In this paper we study special decompositions of a number which are closely related to distance Fibonacci numbers , for .

Let be a fixed integer. A decomposition of the number of the form (resp., ) where , is called an -decomposition (resp., -decomposition). We denote the number of all -decompositions (resp., -decompositions) by (resp., ). Clearly

A decomposition of the number of the form , where , , is called a -decomposition. We denote the number of all -decompositions by .

Let be a fixed integer. A decomposition of the number of the form (resp., ) where , is called an -decomposition (resp., -decomposition). Consequently as the above we denote the number of all -decompositions (resp., -decompositions) by (resp., ). Clearly for a -decomposition of is a -decomposition and for an -decomposition (resp., -decompositions) is an -decompositions (resp., -decompositions). From the above definitions immediately follow relations between numbers , , and (resp., , , and ):

Theorem 1 (see [1]). Let , be integers. Then .

For the proof of the next theorem we will need the following lemma.

Lemma 2. Let , be integers. Then

Proof. If , then the equality immediately follows. Assume that the lemma is true for an arbitrary and we prove it for . Using the definitions of numbers and we obtain that = by the induction’s hypothesis.

We can write the above lemma also in the following form.

Corollary 3. Let , be integers. Then

Theorem 4. Let , , be integers. Then(i),(ii),(iii).

Proof. The equality (i) follows immediately by Theorem 1 and (4).
We shall show that . If , then there is no -decomposition of the number into parts and . So . If , then there is a unique -decomposition of the number ; hence . Let . Assume that the equality holds for an arbitrary . We shall show that . Let be a -decomposition of the number into parts and . If , then so . By induction’s hypothesis there are -decompositions in this case. If then proving analogously we obtain -decompositions of the form . From the above we have -decompositions of the number into parts and , and by the definition of it follows that .
Now we shall prove that . From the definition of we obtain that by the statements (i) and (ii) of this theorem.
Then the statement (iii) follows immediately by Lemma 2, which ends the proof.

Theorem 5. Let , and be integers. Then for natural and .

Proof. Let be natural and . The number is equal to , for , and all -decompositions of have the form . We can put on positions, so we have possibilities. The sum is equal to which ends the proof.

Now we give applications of distance Fibonacci numbers for counting of the number of other special decompositions of the number .

Let , be integers and let be the numbers of all decomposition of the number , where , for and .

Theorem 6. Let , be integers. Then

Proof. Let be a decomposition of the number , where , for and . Then , where is either a -decomposition or a -decomposition of the number . Since by Theorem 4(i) it follows that , which ends the proof.

Theorem 7. Let , be integers. Then for natural and .

Proof. Let , be integers. Let be a decomposition of the number , where , for and . Then the number of such decomposition is equal to the number of -decomposition of the number . By Theorem 4(ii) we obtain that we have decompositions . Since it is clear that totally we have decompositions of the number .
On the other hand, we have that the total number of decompositions is equal to the number of -decompositions of the number . Since by formula (5) and Theorem 4 we obtain From the above it immediately follows that which ends the proof.

3. Identities for , , and

In this section we give some identities for distance Fibonacci numbers: , for .

Theorem 8. Let , be integers. Then

Proof (by induction on ). For we have and
Let . Assume that (17) is true for arbitrary and we prove it for . Using induction’s assumption for and we have which ends the proof.

Corollary 9. Let , be integers. Then(iv),(v),(vi).

Proof. (iv) This formula directly follows from Theorem 8 putting and . For (v) and (vi) analogously.

Theorem 10. Let , be integers. Then

Proof (by induction on ). For we have . The right side of (20) has the form + + .
Let . Assume that (20) is true for arbitrary and we prove it for . Using induction’s assumption for and we have which ends the proof.

4. Matrix Generators and Combinatorial Formulas for , , and

Matrix methods are important in recurrence relations. In the last decades some mathematicians have studied to find miscellaneous affinities between matrices and linear recurrences. Using matrix methods different identities and algebraic representations of considered sequences can be obtained, for instance [1, 2, 46]. Theory of Fibonacci numbers was previously complemented by the theory of so-called the Fibonacci -matrix or the golden matrix. It is worth mentioning that the American mathematician V. Hoggat was one of the first mathematicians who paid the attention to the -matrix. For the classical Fibonacci sequence the matrix generators, named as the golden matrix, have the form and , for .

Golden number and golden section have many interesting applications in different areas of science (physics, chemistry, and mechanics); see for example [7, 8]. In this section we give the matrix generators for distance Fibonacci numbers , where . The matrix generator of the distance Fibonacci numbers of the first kind was introduced in [1] and in this paper we apply this method for all kinds of distance Fibonacci numbers.

Let be a fixed integer. Let be a square matrix of size . For a fixed an element is equal to the coefficient of in the recurrence formula for the distance Fibonacci numbers , . For we define as follows:

In other words, and the matrix is named as the distance Fibonacci matrix or the generator of the distance Fibonacci numbers , .

For a fixed we define the square matrix of size named as the matrix of initial conditions of the formwhere

Theorem 11. Let , be integer. Then for a fixed holds

Since the proof is analogous as in [1] we omit it. To obtain other matrix generators apart distance Fibonacci numbers , , we define a collection of special sequences which are given by the same th order linear recurrence relations as , . These sequences give auxiliary tools for other matrix generators of , and their explicit formulas.

Let , be integers. Let , where is the sequence defined as follows: with the initial conditions and for .

The number will be also denoted shortly by and named as the th distance Fibonacci number of the fourth kind. If and , then .

By simple observation we obtain the following relations between numbers , for :

Using sequences from the collection we can generate the distance Fibonacci numbers , :

Theorem 12. Let be integer. Then(i) for ,(ii) for ,(iii) + for .

Proof. Let denote a sequence defined by the same recurrence as with initial conditions Then We prove analogously formulas (ii) and (iii).

Using the above theorem we obtain a new matrix generator for distance Fibonacci numbers , .

Corollary 13. For the distance Fibonacci numbers , , Theorem 12 gives its matrix generator, respectively:(1) for ,(2) for ,(3) for .

Theorem 14. Let , be integer. Then

Proof. We consider a digraph represented by adjacency matrix auxiliary (Figure 1).
Note that matrix has the following form:
It is well known that is equal to the number of all distinct paths of length between vertices and in the digraph .
Each path from to in digraph has the following form: where denotes the unique shortest path from to in digraph . Parts are cycles of the length , where is or . Thus the length of the path is equal to . There exists one-to-one correspondence between the path and a tuple . If the path has a length , then the corresponding tuple is a decomposition of an integer when occurs times and occurs times and . The number of such tuples is equal to the binomial coefficient . We determine analogously the number of different paths between other pairs of vertices.

491591.fig.001
Figure 1: A digraph .

Using Theorems 14 and 12 we can prove the following.

Theorem 15. Let be integer. Then

Tables 1, 2, 3, and 4 present first words of four types of distance Fibonacci numbers , , , and .

tab1
Table 1: The th distance Fibonacci numbers .
tab2
Table 2: The th distance Fibonacci numbers .
tab3
Table 3: The th distance Fibonacci numbers .
tab4
Table 4: The th distance Fibonacci numbers .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referee for helpful valuable suggestions which resulted in improvements to this paper.

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