On 𝑘-Distance Pell Numbers in 3-Edge-Coloured Graphs

Abstract

We define in this paper new distance generalizations of the Pell numbers and the companion Pell numbers. We give a graph interpretation of these numbers with respect to a special 3-edge colouring of the graph.

1. Introduction

The Fibonacci sequence is defined by the following recurrence relation 𝐹_𝑛=𝐹_𝑛−1+𝐹_𝑛−2 for 𝑛≥2 with 𝐹₀=𝐹₁=1. Among sequences of the Fibonacci type there is the Pell sequence defined by 𝑃_𝑛=2𝑃_𝑛−1+𝑃_𝑛−2 for 𝑛≥2 with the initial conditions 𝑃₀=0 and 𝑃₁=1. The companion Pell sequence is closely related to the Pell sequence and is given by formula 𝑄₀=𝑄₁=2 and 𝑄_𝑛=2𝑄_𝑛−1+𝑄_𝑛−2 for 𝑛≥2. The Pell sequences play an important role in the number theory and they have many interesting interpretations. We recall some of them.(i)The number of lattice paths from the point 𝑂(0,0) to the line 𝑥=𝑛 consisting of 𝑎=[1,1], 𝑏=[1,−1], and 𝑐=[2,0] steps is equal to 𝑃_𝑛+1; see [1].(ii)The number of compositions (i.e., ordered partitions) of a number 𝑛 into two sorts of of 1’s and one sort of 2’s is equal to 𝑃_𝑛+1, see [1].(iii)The number of 𝑛-step non-self-intersecting paths starting at the point 𝑂(0,0) with steps of types 𝑝=[1,0], 𝑞=[−1,0], or 𝑟=[0,1] is equal to (1/2)𝑄_𝑛+1, see [1].

Interesting generalizations of the numbers of the Fibonacci type (also generalizations of the Pell sequences) are studied, for instance, by Kilic in [2–4]. Among others Kilic in [4] introduced the generalization of the Pell numbers. He defined the generalized Pell (𝑝,𝑖)-numbers for any given 𝑝, where 𝑝≥1 and for 𝑛>𝑝+1 and 0≤𝑖≤𝑝 in the following way 𝑃_𝑝^(𝑖)(𝑛)=2𝑃_𝑝^(𝑖)(𝑛−1)+𝑃_𝑝^(𝑖)(𝑛−𝑝−1) with initial conditions and 𝑃_𝑝^(𝑖)(1)=⋯=𝑃_𝑝^(𝑖)(𝑖)=0,  𝑃_𝑝^(𝑖)(𝑖+1)=⋯=𝑃_𝑝^(𝑖)(𝑝+1)=1. If 𝑖=0 then 𝑃_𝑝^(𝑖)(1)=⋯=𝑃_𝑝^(𝑖)(𝑝+1)=1.

In this paper we describe new kinds of generalized Pell sequence and the companion Pell sequence. Our generalization is closely related to the recurrence given in [4] by Kilic. By other initial conditions we obtain other generalized Pell sequences. We give their graph interpretations which are closely related to a concept of edge colouring in graphs. Graph interpretations of the Fibonacci numbers and the like are study intensively; see, for example, [5–9].

2. 𝑘-Distance Pell Sequences 𝑃_𝑘(𝑛) and 𝑄_𝑘(𝑛)

Let 𝑘≥2, 𝑛≥0 be integers. The 𝑘-distance Pell sequence 𝑃_𝑘(𝑛) is defined by the 𝑘th order linear recurrence relation: 𝑃_𝑘(𝑛)=2𝑃_𝑘(𝑛−1)+𝑃_𝑘(𝑛−𝑘)for𝑛≥𝑘, (1) with the following initial conditions: 𝑃_𝑘(𝑛)=0if2𝑛=0,_𝑛−1if𝑛=1,2,3,…,𝑘−1. (2) If 𝑘=2, then this definition reduces to the classical Pell numbers; that is, 𝑃₂(𝑛)=𝑃_𝑛.

Table 1 includes a few first words of the 𝑃_𝑘(𝑛) for special values of 𝑘.

Table 1 

The 𝑘-distance Pell sequence 𝑃_𝑘(𝑛).

Firstly we give an interpretation of the 𝑘-distance Pell sequence, which generalizes result given in (i).

Theorem 1. Let 𝑘≥2 and 𝑛≥1 be integers. Then the number of lattice paths from the point 𝑂(0,0) to the line 𝑥=𝑛 consisting of 𝑎=[1,1], 𝑏=[1,−1], and 𝑐=[𝑘,0] steps is equal to 𝑃_𝑘(𝑛+1).

Proof. Let 𝑠(𝑘,𝑛) be the number of paths from the point 𝑂(0,0) to the line 𝑥=𝑛. It is easy to notice that 𝑠(𝑘,𝑛)=𝑃_𝑘(𝑛+1) for 𝑛=1,2,…,𝑘.
Let 𝑛≥𝑘+1 and let 𝑠_𝑎(𝑘,𝑛), 𝑠_𝑏(𝑘,𝑛), and 𝑠_𝑐(𝑘,𝑛) denote the number of such paths from the point 𝑂(0,0) to the line 𝑥=𝑛, for which the last step is of the form 𝑎=[1,1], 𝑏=[1,−1], and 𝑐=[𝑘,0], respectively.
It can be easily seen that 𝑠_𝑎(𝑘,𝑛)=𝑠(𝑘,𝑛−1), 𝑠_𝑏(𝑘,𝑛)=𝑠(𝑘,𝑛−1), and 𝑠_𝑐(𝑘,𝑛)=𝑠(𝑘,𝑛−𝑘). Since 𝑠(𝑘,𝑛)=𝑠_𝑎(𝑘,𝑛)+𝑠_𝑏(𝑘,𝑛)+𝑠_𝑐(𝑘,𝑛) then we have 𝑠(𝑘,𝑛)=2𝑠(𝑘,𝑛−1)+𝑠(𝑘,𝑛−𝑘) and consequently 𝑠(𝑘,𝑛)=𝑃_𝑘(𝑛+1), for all 𝑛≥1.

In the same way we can prove the generalization of the result (ii).

Theorem 2. Let 𝑘≥2 and 𝑛≥1 be integers. Then the number of all compositions of the number 𝑛 into two sorts of 1’s and one sort of 𝑘’s is equal to 𝑃_𝑘(𝑛+1).

By analogy to the Pell sequence we introduce a generalization of the companion Pell sequence which generalizes the classical companion Pell sequence in the distance sense.

Let 𝑘≥2, 𝑛≥0 be integers. The 𝑘-distance companion Pell sequence 𝑄_𝑘(𝑛) is defined by the 𝑘th order linear recurrence relation: 𝑄_𝑘(𝑛)=2𝑄_𝑘(𝑛−1)+𝑄_𝑘(𝑛−𝑘)for𝑛≥𝑘, (3) with the initial conditions 𝑄_𝑘(𝑛)=𝑘if2𝑛=0,_𝑛if𝑛=1,2,3,…,𝑘−1. (4) If 𝑘=2 then 𝑄₂(𝑛) gives the classical companion Pell numbers 𝑄_𝑛; that is, 𝑄₂(𝑛)=𝑄_𝑛.

Table 2 includes a few first words of the 𝑄_𝑘(𝑛) for special values of 𝑘.

Table 2 

The 𝑘-distance companion Pell sequence 𝑄_𝑘(𝑛).

The following theorem gives the basic relation between 𝑃_𝑘(𝑛) and 𝑄_𝑘(𝑛).

Theorem 3. Let 𝑘≥2 and 𝑛≥𝑘−1 be integers. Then 𝑄_𝑘(𝑛)=2𝑃_𝑘(𝑛)+𝑘𝑃_𝑘(𝑛−𝑘+1). (5)

Proof (by induction on 𝑛). For 𝑛=𝑘−1 the result follows immediately by the definitions of 𝑃_𝑘(𝑛) and 𝑄_𝑘(𝑛). Let 𝑛≥𝑘. Assume that formula (5) is true for 𝑡=𝑘, 𝑘+1,…,𝑛. We will prove that 𝑄_𝑘(𝑛+1)=2𝑃_𝑘(𝑛+1)+𝑘𝑃_𝑘(𝑛−𝑘+2).
By the induction hypothesis and the definitions of 𝑃_𝑘(𝑛) and 𝑄_𝑘(𝑛), we have that 2𝑃_𝑘(𝑛+1)+𝑘𝑃_𝑘(𝑛−𝑘+2)=22𝑃_𝑘(𝑛)+𝑃_𝑘(𝑛−𝑘+1)+𝑘2𝑃_𝑘(𝑛−𝑘+1)+𝑃_𝑘(𝑛−2𝑘+2)=22𝑃_𝑘(𝑛)+𝑘𝑃_𝑘(𝑛−𝑘+1)+2𝑃_𝑘(𝑛−𝑘+1)+𝑘𝑃_𝑘(𝑛−2𝑘+2)=2𝑄_𝑘(𝑛)+𝑄_𝑘(𝑛−𝑘+1)=𝑄_𝑘(𝑛+1), (6) which ends the proof.

For 𝑘=2 Theorem 3 gives the well-known relation between the classical Pell numbers and the companion Pell numbers; namely, 𝑄_𝑛=2(𝑃_𝑛+𝑃_𝑛−1).

Theorem 4. Let 𝑘≥2, 𝑚≥1, and 𝑛≥0 be integers. Then 𝑃_𝑘(𝑛)+2^𝑚_𝑖=1𝑃_𝑘(𝑛+𝑘𝑖−1)=𝑃_𝑘(𝑛+𝑘𝑚),𝑄_𝑘(𝑛)+2^𝑚_𝑖=1𝑄_𝑘(𝑛+𝑘𝑖−1)=𝑄_𝑘(𝑛+𝑘𝑚). (7)(8)

Proof of (7) (by induction on 𝑚). If 𝑚=1 then the equation is obvious. Let 𝑚≥2. Assume that formula (7) is true for an arbitrary 𝑚. We will prove that 𝑃_𝑘(𝑛)+2∑^𝑚+1_𝑖=1𝑃_𝑘(𝑛+𝑖𝑘−1)=𝑃_𝑘(𝑛+(𝑚+1)𝑘).
By the induction hypothesis and the definition of 𝑃_𝑘(𝑛) we have 𝑃_𝑘(𝑛)+2^𝑚+1_𝑖=1𝑃_𝑘(𝑛+𝑖𝑘−1)=𝑃_𝑘(𝑛)+2^𝑚_𝑖=1𝑃_𝑘(𝑛+𝑖𝑘−1)+2𝑃_𝑘(𝑛+(𝑚+1)𝑘−1)=𝑃_𝑘(𝑛+𝑚𝑘)+2𝑃_𝑘(𝑛+(𝑚+1)𝑘−1)=𝑃_𝑘(𝑛+(𝑚+1)𝑘), (9) which ends the proof of (7).
In the same way we can prove the equality (8), so we omit the proof.

If 𝑛=0 or 𝑛=1, respectively, then from Theorem 4 the following follows.

Corollary 5. Let 𝑘≥2 and 𝑚≥1 be integers. Then 2^𝑚_𝑖=1𝑃_𝑘(𝑘𝑖−1)=𝑃_𝑘(),2𝑚𝑘_𝑚𝑖=1𝑃_𝑘(𝑘𝑖)=𝑃_𝑘()2𝑚𝑘+1−1,_𝑚𝑖=1𝑄_𝑘(𝑘𝑖−1)=𝑄_𝑘()2𝑚𝑘−𝑘,_𝑚𝑖=1𝑄_𝑘(𝑘𝑖)=𝑄_𝑘(𝑚𝑘+1)−2. (10)

For 𝑘=2 we obtain the well-known identities for the classical Pell numbers and the companion Pell numbers; namely, 2^𝑚_𝑖=1𝑃_2𝑖−1=𝑃_2𝑚,2^𝑚_𝑖=1𝑃_2𝑖=𝑃_2𝑚+12−1,_𝑚𝑖=1𝑄_2𝑖−1=𝑄_2𝑚2−2,_𝑚𝑖=1𝑄_2𝑖=𝑄_2𝑚+1−2. (11)

Theorem 6. Let 𝑘≥2 and 𝑚=0,1,2,…,𝑘. Then 𝑃_𝑘(𝑘+𝑚)=2^{𝑘+𝑚−1}+𝑚2^𝑚−1,𝑄_𝑘(𝑘+𝑚)=2^𝑘+𝑚+1+𝑘2^𝑚+𝑚2^𝑚. (12)(13)

Proof of (12) (by induction on 𝑚). If 𝑚=0 then the result is obvious. Assume that 𝑃_𝑘(𝑘+𝑚)=2^{𝑘+𝑚−1}+𝑚2^𝑚−1 for an arbitrary 𝑚, such that 1≤𝑚≤𝑘−1. We will prove that 𝑃_𝑘(𝑘+𝑚+1)=2^𝑘+𝑚+(𝑚+1)2^𝑚.
By the induction hypothesis and the definition of 𝑃_𝑘(𝑛), we have 𝑃_𝑘(𝑘+𝑚+1)=2𝑃_𝑘(𝑘+𝑚)+𝑃_𝑘()2𝑚+1=2^{𝑘+𝑚−1}+𝑚2^𝑚−1+2^𝑚=2^𝑘+𝑚+(𝑚+1)2^𝑚, (14) which ends the proof.
In the same way we can prove the equality (13).

If 𝑚=0, 𝑚=1, or 𝑚=𝑘−1, respectively, then from Theorem 6 we obtain the following corollary.

Corollary 7. Let 𝑘≥2. Then 𝑃_𝑘(𝑘)=2^𝑘−1,𝑃_𝑘(𝑘+1)=2^𝑘𝑃+1,_𝑘(2𝑘−1)=2^𝑘−22^𝑘,𝑄+𝑘−1_𝑘(𝑘)=2^𝑘+1𝑄+𝑘,_𝑘(𝑘+1)=2^𝑘+2𝑄+2𝑘+2,_𝑘(2𝑘−1)=2^2𝑘+2^𝑘−1(2𝑘−1). (15)

Theorem 8. Let 𝑘≥2 and 𝑛≥2𝑘−1 be integers. Then 𝑃_𝑘(𝑛)=2^𝑘+1𝑃_𝑘(𝑛−𝑘)+^𝑘−1_𝑖=12^𝑖𝑃_𝑘(𝑛−𝑘−𝑖),𝑄_𝑘(𝑛)=2^𝑘+1𝑄_𝑘(𝑛−𝑘)+^𝑘−1_𝑖=12^𝑖𝑄_𝑘(𝑛−𝑘−𝑖). (16)(17)

Proof of (16) (by induction on 𝑛). For 𝑛=2𝑘−1 we have the equation 𝑃_𝑘(2𝑘−1)=2^𝑘+1𝑃_𝑘(𝑘−1)+^𝑘−1_𝑖=12^𝑖𝑃_𝑘(𝑘−1−𝑖). (18) By the initial conditions of the sequence 𝑃_𝑘(𝑛) and Theorem 6 we can see that this equation is an identity (because it is equivalent to third identity from Corollary 7).
Let 𝑛≥2𝑘. Assume that formula (16) is true for an arbitrary 𝑡 where 2𝑘≤𝑡≤𝑛. We will prove that 𝑃_𝑘(𝑛+1)=(2^𝑘+1)𝑃_𝑘(𝑛+1−𝑘)+∑^𝑘−1_𝑖=12^𝑖𝑃_𝑘(𝑛+1−𝑘−𝑖). By the induction hypothesis and the definition of 𝑃_𝑘(𝑛), we have 2^𝑘+1𝑃_𝑘(𝑛+1−𝑘)+^𝑘−1_𝑖=12^𝑖𝑃_𝑘()=2𝑛+1−𝑘−𝑖_𝑘+12𝑃_𝑘(𝑛−𝑘)+𝑃_𝑘(+𝑛+1−2𝑘)_{𝑘−1𝑖=1}2^𝑖2𝑃_𝑘(𝑛−𝑘−𝑖)+𝑃_𝑘(2𝑛+1−2𝑘−𝑖)=2^𝑘+1𝑃_𝑘(𝑛−𝑘)+^𝑘−1_𝑖=12^𝑖𝑃_𝑘(+2𝑛−𝑘−𝑖)_𝑘+1𝑃_𝑘()+𝑛−2𝑘+1_{𝑘−1𝑖=1}2^𝑖𝑃_𝑘(𝑛+1−2𝑘−𝑖)=2𝑃_𝑘(𝑛)+𝑃_𝑘(𝑛+1−𝑘)=𝑃_𝑘(𝑛+1), (19) which ends the proof of (16). In the same way we can prove (17) which completes the proof of the theorem.

If 𝑘=2 then from Theorem 8 we obtain the following formula for the classical Pell numbers and companion Pell numbers: 𝑃_𝑛=5𝑃_𝑛−2+2𝑃_𝑛−3,𝑄_𝑛=5𝑄_𝑛−2+2𝑄_𝑛−3,𝑛≥3. (20)

3. (𝐴,𝐵,𝑘𝐶)-Coloured Graphs

For concepts not defined here see [10]. The numbers of the Fibonacci type have many applications in distinct areas of mathematics. There is a large interest of modern science in the applications of the numbers of the Fibonacci type. These numbers are studied intensively in a wide sense also in graphs and combinatorials problem. In graphs Prodinger and Tichy initiated studying the Fibonacci numbers and the like. In [11] they showed the relations between the number of independent sets in 𝒫_𝑛 and 𝒞_𝑛 with the Fibonacci numbers and the Lucas numbers, where 𝒫_𝑛 and 𝒞_𝑛 denote an 𝑛-vertex path and an 𝑛-vertex cycle, respectively. This short paper gave an impetus for counting problems related to the numbers of the Fibonacci type. Many of these problems and results are closely related with the Merrifield-Simmons index 𝜎(𝐺) and the Hosoya index 𝑍(𝐺) in graphs; see [6, 12]. The Pell numbers also have a graph interpretation. It is well-known that 𝑍(𝒫_𝑛∘𝐾₁)=𝑃_𝑛+1, where 𝐺∘𝐻 denotes the corona of two graphs.

In this section we give a graph interpretation of the 𝑘-distance Pell numbers with respect to special edge colouring of a graph.

Let 𝐺 be a 3-edge coloured graph with the set of colours {𝐴,𝐵,𝐶}. Let 𝑀∈{𝐴,𝐵,𝐶}. We say that a path is 𝑀-monochromatic if all its edges are coloured alike by colour 𝑀. By 𝑙(𝑀) we denote the length of the 𝑀-monochromatic path. For 𝑥𝑦∈𝐸(𝐺) notation 𝑀(𝑥𝑦) means that the edge 𝑥𝑦 has the colour 𝑀.

Let 𝑘≥1 be an integer. In the graph 𝐺 we define a (𝐴,𝐵,𝑘𝐶)-edge colouring, such that 𝑙(𝐴)≥0, 𝑙(𝐵)≥0, and 𝑙(𝐶)=𝑞𝑘, where 𝑞≥0 is an integer.

Theorem 9. Let 𝑘≥2 and 𝑛≥2 be integers. The number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛 is equal to 𝑃_𝑘(𝑛).

Proof. Let 𝑉(𝒫_𝑛)={𝑥₁,𝑥₂,…,𝑥_𝑛} be the set of vertices of a graph 𝒫_𝑛 with the numbering in the natural fashion. Then 𝐸(𝒫_𝑛)={𝑥₁𝑥₂,𝑥₂𝑥₃,…,𝑥_𝑛−1𝑥_𝑛}. Let 𝜎(𝑘,𝑛) be the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛. By inspection we obtain that 𝜎(𝑘,𝑛)=𝑃_𝑘(𝑛) for 𝑛=2,…,𝑘+1.
Let 𝑛≥𝑘+2 and let 𝜎_𝐴(𝑘,𝑛), 𝜎_𝐵(𝑘,𝑛), and 𝜎_𝐶(𝑘,𝑛) denote the number of (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛, with 𝐴(𝑥_𝑛−1𝑥_𝑛), 𝐵(𝑥_𝑛−1𝑥_𝑛), and 𝐶(𝑥_𝑛−1𝑥_𝑛), respectively. It is obvious that 𝜎_𝐴(𝑘,𝑛)=𝜎_𝐵(𝑘,𝑛).
Clearly 𝜎_𝐴(𝑘,𝑛) and 𝜎_𝐵(𝑘,𝑛) are equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛−1 and 𝜎_𝐶(𝑘,𝑛) is equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛−𝑘. In the other words 𝜎_𝐴(𝑘,𝑛)=𝜎(𝑘,𝑛−1), 𝜎_𝐵(𝑘,𝑛)=𝜎(𝑘,𝑛−1), and 𝜎_𝐶(𝑘,𝑛)=𝜎(𝑘,𝑛−𝑘). Since 𝜎(𝑘,𝑛)=𝜎_𝐴(𝑘,𝑛)+𝜎_𝐵(𝑘,𝑛)+𝜎_𝐶(𝑘,𝑛), (21) then we have 𝜎(𝑘,𝑛)=2𝜎(𝑘,𝑛−1)+𝜎(𝑘,𝑛−𝑘). By the initial conditions we have that 𝜎(𝑘,𝑛)=𝑃_𝑘(𝑛), for all 𝑛≥2, which ends the proof.

Corollary 10. Let 𝑛≥2 be integer. The number of all (𝐴,𝐵,2𝐶)-edge colouring of the graph 𝒫_𝑛 is equal to 𝑃_𝑛.

Using the concept of (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛 we can obtain the direct formula for the numbers 𝑃_𝑘(𝑛) and 𝑄_𝑘(𝑛).

Let 𝑘≥1, 𝑛≥2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers and let 𝑝_𝑘(𝑛,𝑡) be the number of (𝐴,𝐵,𝑘𝐶)-edge colouring of graph 𝒫_𝑛, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡 times. In the other words 𝑡𝑘 edges of 𝒫_𝑛 have colour 𝐶.

Theorem 11. Let 𝑛≥2 and 0≤𝑡≤𝑛−1 be integers. Then 𝑝₁(𝑛,𝑡)=_𝑡^𝑛−12^{𝑛−1−𝑡}.

Proof. Since 𝒫_𝑛 has 𝑛−1 edges, then for 𝑡=0,1,2,… we obtain that 𝑝₁(𝑛,0)=2^𝑛−1,𝑝₁()=1𝑛,12𝑛−1_𝑛−2,𝑝₁()=2𝑛,22𝑛−1_𝑛−3,…, (22) and so 𝑝₁(𝑛,𝑡)=_𝑡^𝑛−12^{𝑛−1−𝑡}.

Theorem 12. Let 𝑘≥1, 𝑛≥2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then for all 𝑠=0,1,2,…,𝑘−1 holds 𝑝_𝑘(𝑛,𝑡)=𝑝_𝑘−𝑠(𝑛−𝑡𝑠,𝑡).

Proof. Consider the graph 𝒫_𝑛 with the (𝐴,𝐵,𝑘𝐶)-edge colouring. By contracting the 𝐶-monochromatic paths of length 𝑘 to the 𝐶-monochromatic paths of length 𝑘−𝑠 we can see that the number of (𝐴,𝐵,𝑘𝐶)-edge colouring of graph 𝒫_𝑛, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡 times is equal to the number of (𝐴,𝐵,(𝑘−𝑠)𝐶)-edge colouring of graph 𝒫_{𝑛−𝑡𝑠}, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡 times and the Theorem follows.

If 𝑠=𝑘−1 or 𝑠=𝑘−2 then Theorem 12 gives the following result.

Corollary 13. Let 𝑘≥1, 𝑛≥2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then 𝑝_𝑘(𝑛,𝑡)=𝑝₁(𝑛−𝑡(𝑘−1),𝑡),𝑝_𝑘(𝑛,𝑡)=𝑝₂(𝑛−𝑡(𝑘−2),𝑡). (23)(24)

By (23) and Theorem 11 we obtain the direct formula for 𝑝_𝑘(𝑛,𝑡).

Corollary 14. Let 𝑘≥1, 𝑛≥2, 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then 𝑝_𝑘()=()𝑡𝑡𝑛,𝑡2𝑛−1−𝑘−1_{𝑛−1−𝑘𝑡}. (25)

Moreover Theorem 9 immediately gives that _𝑡≥0𝑝_𝑘(𝑛,𝑡)=𝜎(𝑘,𝑛)=𝑃_𝑘(𝑛). (26)

Using (26) and Corollary 14 we can give the direct formula for the number 𝑃_𝑘(𝑛).

Theorem 15. Let 𝑘≥2 and 𝑛≥1 be integers. Then 𝑃_𝑘(𝑛)=^{[(𝑛−1)/𝑘]}_𝑖=0()𝑖𝑖2𝑛−1−𝑘−1_{𝑛−1−𝑘𝑖}. (27)

If 𝑘=2 then we obtain the direct formula for the classical Pell numbers of the form 𝑃_𝑛=^{[(𝑛−1)/2]}_𝑖=0𝑖2𝑛−1−𝑖_{𝑛−1−2𝑖}=^{[(𝑛−1)/2]}_𝑖=0𝑛22𝑖+1^𝑖. (28)

From Theorems 3 and 15 follows the direct formula for the sequence 𝑄_𝑘(𝑛).

Theorem 16. Let 𝑘≥2 and 𝑛≥1 be integers. Then 𝑄_𝑘(𝑛)=2^{[(𝑛−1)/𝑘]}_𝑖=0()𝑖𝑖2𝑛−1−𝑘−1_{𝑛−1−𝑘𝑖}+𝑘^{[(𝑛−𝑘)/𝑘]}_𝑖=0()𝑖𝑖2𝑛−𝑘−𝑘−1_{𝑛−𝑘−𝑘𝑖}. (29)

If 𝑘=2 then using the above theorem and after some calculations we obtain that 𝑄_𝑛=2^{[(𝑛−1)/2]}_𝑖=0𝑖2𝑛−1−𝑖_{𝑛−1−2𝑖}+^{[(𝑛−2)/2]}_𝑖=0𝑖2𝑛−2−𝑖_{𝑛−2−2𝑖}. (30)

Now we give the recurrence relation for the number 𝑝_𝑘(𝑛,𝑘).

Theorem 17. Let 𝑘≥1, 𝑛≥𝑘+2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then 𝑝_𝑘(𝑛,𝑡)=2𝑝_𝑘(𝑛−1,𝑡)+𝑝_𝑘(𝑛−𝑘,𝑡−1). (31)

Proof. Let 𝑉(𝒫_𝑛)={𝑥₁,𝑥₂,…,𝑥_𝑛} be the set of vertices of a graph 𝒫_𝑛 with the numbering in the natural fashion. Then 𝐸(𝒫_𝑛)={𝑥₁𝑥₂,𝑥₂𝑥₃,…,𝑥_𝑛−1𝑥_𝑛}.
Let 𝑝^𝐴_𝑘(𝑛,𝑡), 𝑝^𝐵_𝑘(𝑛,𝑡), and 𝑝^𝐶_𝑘(𝑛,𝑡) denote the number of (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡 times with 𝐴(𝑥_𝑛−1𝑥_𝑛), 𝐵(𝑥_𝑛−1𝑥_𝑛), and 𝐶(𝑥_𝑛−1𝑥_𝑛), respectively.
It can be easily seen that both 𝑝^𝐴_𝑘(𝑛,𝑡) and 𝑝^𝐵_𝑘(𝑛,𝑡) are equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛−1, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡 times and 𝑝^𝐶_𝑘(𝑛,𝑡) is equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛−𝑘, such that 𝐶-monochromatic path appears in this colouring exactly 𝑡−1 times. In the other words 𝑝^𝐴_𝑘(𝑛,𝑡)=𝑝_𝑘(𝑛−1,𝑡), 𝑝^𝐵_𝑘(𝑛,𝑡)=𝑝_𝑘(𝑛−1,𝑡), and 𝑝^𝐶_𝑘(𝑘,𝑛)=𝑝_𝑘(𝑛−𝑘,𝑡−1). Since 𝑝_𝑘(𝑛,𝑡)=𝑝^𝐴_𝑘(𝑛,𝑡)+𝑝^𝐵_𝑘(𝑛,𝑡)+𝑝^𝐶_𝑘(𝑛,𝑡), (32) so the result immediately follows.

By (24) and by Theorem 17 we obtain the following.

Corollary 18. Let 𝑘≥1, 𝑛≥𝑘+2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then 𝑝_𝑘(𝑛,𝑡)=2𝑝₂(𝑛−𝑡(𝑘−2)−1,𝑡)+𝑝₂(𝑛−𝑡(𝑘−2)−𝑘,𝑡−1). (33)

Using (26) and Corollary 18 we have the following.

Corollary 19. Let 𝑘≥2, 𝑛≥𝑘+2, and 0≤𝑡≤[(𝑛−1)/𝑘] be integers. Then 𝑃_𝑘(𝑛)=^{[(𝑛−1)/𝑘]}_𝑡=02𝑝₂(𝑛−𝑡(𝑘−2)−1,𝑡)+𝑝₂(𝑛−𝑡(𝑘−2)−𝑘,𝑡−1). (34)

Now we consider the (𝐴,𝐵,𝑘𝐶)-edge colouring of the cycle 𝒞_𝑛 with the numbering of edges in the natural fashion. For explanation for cycle 𝒞₄ with 𝐸(𝒞₄)={𝑒₁,𝑒₂,𝑒₃,𝑒₄} there are two distinct (𝐴,𝐵,2𝐶)-edge colouring using only colour 𝐶. The first (𝐴,𝐵,2𝐶)-edge colouring gives a partition of the set 𝐸(𝒞₄)={𝑒₁,𝑒₂}∪{𝑒₃,𝑒₄} and the second (𝐴,𝐵,2𝐶)-edge colouring gives a partition 𝐸(𝒞₄)={𝑒₂,𝑒₃}∪{𝑒₄,𝑒₁}.

Let 𝜌(𝑘,𝑛) be the number of all (𝐴,𝐵,2𝐶)-edge colouring of the cycle 𝒞_𝑛.

Theorem 20. Let 𝑘≥2 and 𝑛≥3 be integers. The number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the cycle 𝒞_𝑛 is equal to 𝑄_𝑘(𝑛).

Proof. Let 𝑉(𝒞_𝑛)={𝑥₁,𝑥₂,…,𝑥_𝑛} be the set of vertices of a graph 𝒞_𝑛 with the numbering in the natural fashion. Then 𝐸(𝒞_𝑛)={𝑥₁𝑥₂,𝑥₂𝑥₃,…,𝑥_𝑛−1𝑥_𝑛,𝑥_𝑛𝑥₁}.
It is easy to check that 𝜌(𝑘,𝑛)=𝑄_𝑘(𝑛) for 𝑛=3,…,𝑘.
Let 𝑛≥𝑘+1 and let 𝜌_𝐴(𝑘,𝑛), 𝜌_𝐵(𝑘,𝑛), and 𝜌_𝐶(𝑘,𝑛) denote the number of (𝐴,𝐵,𝑘𝐶)-edge colouring of the cycle 𝒞_𝑛 with 𝐴(𝑥_𝑛𝑥₁), 𝐵(𝑥_𝑛𝑥₁), and 𝐶(𝑥_𝑛𝑥₁), respectively.
It can be easily seen that 𝜌_𝐴(𝑘,𝑛) and 𝜌_𝐵(𝑘,𝑛) are equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_𝑛 and 𝜌_𝐶(𝑘,𝑛) is equal to the number of all (𝐴,𝐵,𝑘𝐶)-edge colouring of the graph 𝒫_{𝑛−𝑘+1} multiplied by 𝑘. By Theorem 9 we have 𝜌_𝐴(𝑘,𝑛)=𝑃_𝑘(𝑛), 𝜌_𝐵(𝑘,𝑛)=𝑃_𝑘(𝑛), and 𝜌_𝐶(𝑘,𝑛)=𝑘𝑃_𝑘(𝑛−𝑘+1). Since 𝜌(𝑘,𝑛)=𝜌_𝐴(𝑘,𝑛)+𝜌_𝐵(𝑘,𝑛)+𝜌_𝐶(𝑘,𝑛), (35) then we have 𝜌(𝑘,𝑛)=2𝑃_𝑘(𝑛)+𝑘𝑃_𝑘(𝑛−𝑘+1) and so by Theorem 3 we have 𝜌(𝑘,𝑛)=𝑄_𝑘(𝑛) for all 𝑛≥3.

Corollary 21. Let 𝑛≥3 be an integer. Then the number of all (𝐴,𝐵,2𝐶)-edge colouring of the cycle 𝒞_𝑛 is equal to 𝑄_𝑛, 𝑛≥3.

4. Concluding Remarks

The interpretation of the generalized Pell numbers with respect to (𝐴,𝐵,𝑘𝐶)-colouring of a 3-edge coloured graph gives a motivation for studying this type of colouring in graphs. For an arbitrary 𝑘≥2 this problem seems to be difficult and more interesting results can be obtained for special value of 𝑘 (e.g., if we study (𝐴,𝐵,2𝐶)-colouring in graphs). In the class of trees some interesting results can be obtained.

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.