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

p (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].


Introduction
The Fibonacci sequence is defined by the following recurrence relation   =  −1 +  −2 for  ≥ 2 with  0 =  1 = 1.Among sequences of the Fibonacci type there is the Pell sequence defined by   = 2 −1 + −2 for  ≥ 2 with the initial conditions  0 = 0 and  1 = 1.The companion Pell sequence is closely related to the Pell sequence and is given by formula  0 =  1 = 2 and   = 2 −1 +  −2 for  ≥ 2. The Pell sequences play an important role in the number theory and they have many interesting interpretations.We recall some of them.
(ii) The number of compositions (i.e., ordered partitions) of a number  into two sorts of of 1's and one sort of 2's is equal to  +1 , see [1].

𝑘-Distance Pell Sequences 𝑃 𝑘 (𝑛) and 𝑄 𝑘 (𝑛)
Let  ≥ 2,  ≥ 0 be integers.The -distance Pell sequence   () is defined by the th order linear recurrence relation: with the following initial conditions: ( If  = 2, then this definition reduces to the classical Pell numbers; that is,  2 () =   .Table 1 includes a few first words of the   () for special values of .
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  ≥ 2,  ≥ 0 be integers.The -distance companion Pell sequence   () is defined by the th order linear recurrence relation: with the initial conditions If  = 2 then  2 () gives the classical companion Pell numbers   ; that is,  2 () =   .
Table 2 includes a few first words of the   () for special values of .
By the induction hypothesis and the definition of   () we have which ends the proof of (7).
In the same way we can prove the equality ( 8), so we omit the proof.
In the same way we can prove the equality (13).
(15) Theorem 8. Let  ≥ 2 and  ≥ 2 − 1 be integers.Then Proof of (16) (by induction on ).For  = 2 − 1 we have the equation By the initial conditions of the sequence   () and Theorem 6 we can see that this equation is an identity (because it is equivalent to third identity from Corollary 7).
If  = 2 then from Theorem 8 we obtain the following formula for the classical Pell numbers and companion Pell numbers: (20)

(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  and C  with the Fibonacci numbers and the Lucas numbers, where P  and C  denote an -vertex path and an -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 () and the Hosoya index () in graphs; see [6,12].The Pell numbers also have a graph interpretation.It is well-known that (P  ∘  1 ) =  +1 , where  ∘  denotes the corona of two graphs.
In this section we give a graph interpretation of the distance Pell numbers with respect to special edge colouring of a graph.
Let  be a 3-edge coloured graph with the set of colours {, , }.Let  ∈ {,,}.We say that a path is monochromatic if all its edges are coloured alike by colour .By () we denote the length of the -monochromatic path.For  ∈ () notation () means that the edge  has the colour .
Corollary 10.Let  ≥ 2 be integer.The number of all (, , 2)-edge colouring of the graph P  is equal to   .
Using the concept of (, , )-edge colouring of the graph P  we can obtain the direct formula for the numbers   () and   ().
Corollary 21.Let  ≥ 3 be an integer.Then the number of all (, , 2)-edge colouring of the cycle C  is equal to   ,  ≥ 3.

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