On Certain Fourth-Order Linear Recursive Sequences

Abstract

In this paper, we introduce a fourth-order linear recursive sequence that is related to the concept of subbalancing numbers. This sequence is constructed by using the third balancing number in the Diophantine equation of subbalancing numbers and is called the sequence of $B_3$-Lucas subbalancing numbers. Motivated by the results for $b_3$-Lucas subbalancing numbers, we obtain several algebraic identities regarding $B_3$-Lucas subbalancing numbers. Furthermore, we derive some algebraic relations between $B_3$-Lucas subbalancing numbers and some other integer sequences.

1. Introduction

Symmetry plays various significant roles in all branches of mathematics. Some characteristic examples regarding this fact are illustrated in [1]. Many studies have been conducted regarding symmetry in number theory, especially recursive sequences and related polynomials [2,3,4,5,6]. In number theory, the term ‘recursive sequence’ refers to a sequence that satisfies a recurrence relation. As each term in these kinds of sequences is defined as a function of the terms that came before it, a recurrence relation is an equation that characterizes a sequence recursively. The subject of recursive sequences is obviously of fundamental relevance and has long been a key component of number theory. These sequences are found almost everywhere in mathematics and computer science. Examples include the theory of power series for rational functions [7], cellular automata [8], k-regular [9], automated sequences [10], and pseudo-random number generators [11,12]. Several surveys regarding the properties of linear recurrence sequences can be found in the literature; see for example [13,14,15,16,17,18].

One of the most studied classes of recursive sequences is the class of linear recursive sequences. In [19], the sequence $\left\{G_n\right\}$ satisfying $G_n=A_0{G}_{n-1}+A_1{G}_{n-2}+\cdots +A_{k-1}{G}_{n-k}$ with real numbers $A_0,A_1,\cdots ,A_{k-1}$ ($A_{k-1}\ne 0$) ($n\ge k$) is called a linear recursive sequence, where $k\ge 2$ is a fixed number. The integer k is called the order of linear recurrence. Linear recursive sequences are names according to these degrees. The kth order linear recursive sequence is defined by its first k terms and the recurrence relation as mentioned.

Two of the most famous of these sequences are Fibonacci and Lucas sequences, which are second order linear recursive sequences. The recurrence relations of Fibonacci and Lucas sequences are $F_n=F_{n-1}+F_{n-2}$ and $L_n=L_{n-1}+L_{n-2}$ with initial terms of $F_0=0,F_1=1$ and $L_0=2,L_1=1$, respectively. In addition to these mentioned sequences, Pell and Pell-Lucas sequences can also be given as examples of second-order linear recursive sequences. The recurrence relations of these sequences are $P_n=2P_{n-1}+P_{n-2}$ and $Q_n=2Q_{n-1}+Q_{n-2}$, with the initial terms $P_1=1,P_2=2$ and $Q_1=1,Q_2=3$, respectively. The applications and identities related to these sequences can be seen in [20,21,22,23,24,25,26,27,28,29,30,31].

Apart from these sequences, there are many linear recursive sequences, and one of them is the sequence of balancing numbers, which forms the basis of our study. The sequence of balancing numbers is a second-order linear recursive sequence and has the recurrence relation $B_n=6B_{n-1}-B_{n-2}$ with the initial terms $B_1=1$ and $B_2=6$, where $B_n$ denotes the nth balancing number [32]. For more information, see Definition 1 on page 3.

Another second-order linear recursive sequence defined as $C_n=\sqrt{8B_n^2+1}$ using balancing numbers is the sequence of Lucas-balancing numbers [33]. This sequence has the recurrence relation $C_n=6C_{n-1}-C_{n-2}$ with the initial terms $C_1=3$ and $C_2=17$ (for further details on these sequences, see [34,35,36,37,38,39,40,41]).

In [42], the definition of the sequence of cobalancing numbers, which is another integer sequence related to the sequence of balancing numbers, is given. The recurrence relation of this sequence is $b_n=6b_{n-1}-b_{n-2}+2$ with the initial terms $b_1=0,b_2=2$, where the nth cobalancing number is denoted by $b_n$. Later in [43], Lucas-cobalancing numbers are defined as $c_n=\sqrt{8b_n^2+8b_n+1}$, similar to the definition of Lucas-balancing numbers. This sequence has the recurrence relation $c_n=6c_{n-1}-c_{n-2}$ with the initial terms $c_1=1$ and $c_2=7$. Note that the sequence of Lucas-cobalancing numbers is a second-order linear recursive sequence.

As well as second-order linear recursive sequences, higher-order linear recursive sequences have attracted the attention of many researchers [44,45,46,47]. One of the most studied classes of higher-order linear recursive sequences is the class of fourth-order linear recursive sequences. The sequences of Tetranacci, Quadrapell, quadra Fibona-Pell, and almost balancing numbers can be given as examples of the fourth-order linear recursive sequences (for more details, see [48,49,50,51,52,53,54,55]).

In this study, we consider a new fourth-order linear recursive sequence that is related to D-subbalancing numbers. In [56], D-subbalancing numbers are introduced by adding a fixed positive integer to the left side of the Diophantine equation that is satisfied by balancing numbers (see Definition 2 on page 3). The choice of the fixed positive integer D is of critical importance in the concept of subbalancing numbers because D-subbalancing numbers cannot be obtained for every value of D. Some studies on the choice of the positive integer D to obtain new subbalancing numbers have been conducted in [56,57,58]. In these studies, it has been shown that D-subbalancing numbers can be obtained when the positive integer D is chosen as the terms of the sequences of cobalancing, balancing, and triangular numbers, respectively (for detailed information regarding the choice of D as the third balancing number, see Theorems 1–4 on page 3). In [59], $b_3$-subbalancing numbers obtained by choosing the positive integer D as the third cobalancing number are examined, and $b_3$-Lucas subbalancing numbers are introduced (see Theorem 5 on page 4).

This study consists of four sections. In Section 1, we provide an introduction for this study. In Section 2, we present the definitions of balancing and subbalancing numbers. Furthermore, we review some results regarding these numbers which will be used in this work. Later in Section 3, we introduce a fourth-order linear recursive sequence that is called $B_3$-Lucas subbalancing numbers. Motivated by the results on $b_3$-Lucas subbalancing numbers, we obtain several algebraic identities regarding $B_3$-Lucas subbalancing numbers. In addition to these, we derive some algebraic relations between $B_3$-Lucas subbalancing numbers and $B_3$-subbalancing numbers and $b_3$-Lucas subbalancing numbers. Finally, we present the conclusions of this work and provide some ideas about subsequent research that can be obtained with the motivation of these conclusions in Section 4.

2. Preliminaries

In this section, we recall some definitions on balancing and subbalancing numbers and provide some auxiliary results that are used in the proof of the main results.

Definition 1

([32]). A positive integer n is called a balancing number if

$$1+2+\cdots +(n-1)=(n+1)+(n+2)+\cdots +(n+r)$$

for some positive integer r. Here, r is called the balancer corresponding to the balancing number n.

As a result of this definition, we can provide the following two corollaries.

Corollary 1.

If n is a balancing number with balancer r, then

$$n^2={\displaystyle \frac{(n+r)(n+r+1)}{2}}andr={\displaystyle \frac{-(2n+1)+\sqrt{8n^2+1}}{2}}.$$

Corollary 2.

n is a balancing number if and only if $8n^2+1$ is a perfect square.

Definition 2

([56]). For a fixed positive integer D, a positive integer n is called a D-subbalancing number if

$$1+2+\cdots +(n-1)+D=(n+1)+(n+2)+\cdots +(n+r)$$

(1)

for some positive integer r. Here, r is called the D-subbalancer corresponding to D-subbalancing number n.

As a result of this definition, we can provide the following two corollaries.

Corollary 3.

If n is a D-subbalancing number with D-subbalancer r, then

$$n^2+D={\displaystyle \frac{(n+r)(n+r+1)}{2}}andr={\displaystyle \frac{-(2n+1)+\sqrt{8n^2+8D+1}}{2}}.$$

Corollary 4.

n is a D-subbalancing number if and only if $8n^2+8D+1$ is a perfect square.

As shown in [57], the positive integer D in the Diophantine Equation (1) can be taken as the terms of the sequence of balancing numbers. By choosing D as the third balancing number, we obtain the sequence of $B_3$-subbalancing numbers. The nth term of the sequence of $B_3$-subbalancing numbers is denoted by ${\left(S{B}_3\right)}_n$.

Theorem 1

([57]). The recurrence relation of the sequence of $B_3$-subbalancing numbers is

$$\begin{array}{c}\hfill {\left(S{B}_3\right)}_n=6{\left(S{B}_3\right)}_{n-2}-{\left(S{B}_3\right)}_{n-4}(n\ge 4),\end{array}$$

where ${\left(S{B}_3\right)}_0=1,{\left(S{B}_3\right)}_1=14,{\left(S{B}_3\right)}_2=20$ and ${\left(S{B}_3\right)}_3=85$.

Corollary 5.

n is a $B_3$-subbalancing number if and only if $8n^2+281$ is a perfect square.

Theorem 2

([57]). Let ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number, $B_m$ denote the mth balancing number, and $C_m$ denote the mth Lucas-balancing number. Then,

$$\begin{array}{ccc}\hfill {\left(S{B}_3\right)}_{2m}& =& 17B_m+C_m,\hfill \\ \hfill {\left(S{B}_3\right)}_{2m+1}& =& 17B_{m+1}-C_{m+1}\hfill \end{array}$$

for $m\ge 0$.

Theorem 3

([57]). Let ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number. Then,

$${\left(S{B}_3\right)}_m^2={\left(S{B}_3\right)}_{m-2}{\left(S{B}_3\right)}_{m+2}+281$$

for $m\ge 2$.

Theorem 4

([57]). Let ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number and $B_m$ denote the mth balancing number. Then,

$${\left(S{B}_3\right)}_{m+1}{\left(S{B}_3\right)}_m-{\left(S{B}_3\right)}_{m-1}{\left(S{B}_3\right)}_{m-2}=281B_m$$

for $m\ge 2$.

Theorem 5

([59]). Let ${\left(CS{b}_3\right)}_m$ denote the mth $b_3$-Lucas subbalancing number, $B_m$ denote the mth balancing number, and $C_m$ denote the mth Lucas-balancing number. Then,

$$\begin{array}{ccc}\hfill {\left(CS{b}_3\right)}_{2m}& =& 11C_m+8B_m,\hfill \\ \hfill {\left(CS{b}_3\right)}_{2m+1}& =& 11C_{m+1}-8B_{m+1}\hfill \end{array}$$

for $m\ge 0$.

In this study, firstly, $B_3$-Lucas subbalancing numbers are introduced using $B_3$-subbalancing numbers. The relationship between balancing and Lucas-balancing numbers led us to investigate a new integer sequence related to $B_3$-subbalancing numbers. It can be observed that $8B_m^2+1$ is a perfect square from Corollary 2. Motivated by this, the sequence of Lucas-balancing numbers was introduced using the square root of $8B_m^2+1$. Similarly, it can be observed that $8{\left(S{B}_3\right)}_m^2+281$ is a perfect square from Corollary 5. The sequence of $B_3$-Lucas subbalancing numbers was thus formed by the successful application of the idea of construction a new integer sequence using the square root of $8{\left(S{B}_3\right)}_m^2+281$. It is observed that some identities similar to those between balancing and Lucas-balancing numbers can also be obtained between the numbers $B_3$-Lucas subbalancing and $B_3$-subbalancing numbers.

The recurrence relation is the notion that is focused on due to its significance in the topic of integer sequences. The recurrence relation of a sequence informs regarding relations between any terms of this sequence and the previous ones. In this context, using the identities that have been established between $B_3$-Lucas subbalancing and $B_3$-subbalancing numbers, the recurrence relation of the sequence of $B_3$-Lucas subbalancing numbers will be determined.

In Theorem 2, the expressions of even and odd indices terms of the sequence of $B_3$-subbalancing numbers depending on the balancing and Lucas-balancing numbers are given. The identities for even and odd indices terms of the sequence of $B_3$-Lucas subbalancing numbers that are connected balancing and Lucas-balancing numbers will be determined using Theorem 2 and the fundamental identities between $B_3$-Lucas subbalancing and $B_3$-subbalancing numbers. In light of the studies examining the relationships between balancing, Lucas-balancing, cobalancing, and Lucas-cobalancing numbers, it is natural to intend searching the relationships between $B_3$-Lucas subbalancing and these numbers. Motivated by this, some algebraic identities regarding the relations of $B_3$-Lucas subbalancing numbers with cobalancing and Lucas-cobalancing numbers will be obtained.

3. Results

In this section, we first introduce the concept of $B_3$-Lucas subbalancing numbers. Then, we deduce some algebraic identities between the sequence of $B_3$-Lucas subbalancing numbers and some other integer sequences. Throughout this paper, we denote the mth $B_3$-Lucas subbalancing number as ${\left(CS{B}_3\right)}_m$.

Definition 3.

Let ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number. The square root of $8{\left(S{B}_3\right)}_m^2+281$ is called the mth $B_3$-Lucas subbalancing number and denoted by ${\left(CS{B}_3\right)}_m$. That is,

$$\begin{array}{c}\hfill {\left(CS{B}_3\right)}_m=\sqrt{8{\left(S{B}_3\right)}_m^2+281}.\end{array}$$

The first few terms of the sequence of $B_3$-Lucas subbalancing numbers are 17, 43, 59, 241, 337.

Corollary 6.

Let ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number and ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number. Then,

$${\left(S{B}_3\right)}_{m-2}=3{\left(S{B}_3\right)}_m-{\left(CS{B}_3\right)}_m(m\ge 2)$$

and

$${\left(S{B}_3\right)}_{m+2}=3{\left(S{B}_3\right)}_m+{\left(CS{B}_3\right)}_m(m\ge 0).$$

Proof. 

It can be proven by using Theorem 3. □

Theorem 6.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number. Then, the recurrence relation of $B_3$-Lucas subbalancing numbers is

$${\left(CS{B}_3\right)}_{m+2}=6{\left(CS{B}_3\right)}_m-{\left(CS{B}_3\right)}_{m-2}(m\ge 2),$$

where ${\left(CS{B}_3\right)}_0=17,{\left(CS{B}_3\right)}_1=43,{\left(CS{B}_3\right)}_2=59$ and ${\left(CS{B}_3\right)}_3=241$.

Proof. 

By using Corollary 6, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{m+2}^2& =& 8{\left(S{B}_3\right)}_{m+2}^2+281\hfill \\ & =& 8{\left[3{\left(S{B}_3\right)}_m+{\left(CS{B}_3\right)}_m\right]}^2+281\hfill \\ & =& {[3{\left(CS{B}_3\right)}_m+8{\left(S{B}_3\right)}_m]}^2.\hfill \end{array}$$

Thus, we obtain

$${\left(CS{B}_3\right)}_{m+2}=3{\left(CS{B}_3\right)}_m+8{\left(S{B}_3\right)}_m.$$

(2)

Similarly, we obtain

$${\left(CS{B}_3\right)}_{m-2}=3{\left(CS{B}_3\right)}_m-8{\left(S{B}_3\right)}_m.$$

(3)

From (2) and (3), we obtain

$${\left(CS{B}_3\right)}_{m+2}=6{\left(CS{B}_3\right)}_m-{\left(CS{B}_3\right)}_{m-2}.$$

□

Theorem 7.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number, $B_m$ denote the mth balancing number, and $C_m$ denote the mth Lucas-balancing number. Then,

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}& =& 17C_m+8B_m,\hfill \\ \hfill {\left(CS{B}_3\right)}_{2m+1}& =& 17C_{m+1}-8B_{m+1}\hfill \end{array}$$

for $m\ge 0$.

Proof. 

By using Theorem 2, Corollary 6, and the relations between balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}& =& {\left(S{B}_3\right)}_{2m+2}-3{\left(S{B}_3\right)}_{2m}\hfill \\ & =& (17B_{m+1}+C_{m+1})-3(17B_m+C_m)\hfill \\ & =& 17(B_{m+1}-3B_m)+(C_{m+1}-3C_m)\hfill \\ & =& 17C_m+8B_m.\hfill \end{array}$$

The other case can be proven similarly. □

This theorem provides us a relation between negative and positive indices terms of the sequence of $B_3$-Lucas subbalancing numbers. As ${\left(CS{B}_3\right)}_{-m}={\left(CS{B}_3\right)}_{m-1}$, this fourth-order linear recursive sequence has a property related to symmetry.

In the following theorem, the Binet formula for $B_3$-Lucas subbalancing numbers is given.

Theorem 8.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number. Then,

$${\left(CS{B}_3\right)}_{2m}={\displaystyle \frac{(68+8\sqrt{2}){\alpha }_1^m+(68-8\sqrt{2}){\alpha }_2^m}{8}},$$

$${\left(CS{b}_3\right)}_{2m+1}={\displaystyle \frac{\left(68-8\sqrt{2}\right){\alpha }_1^{m+1}+\left(68+8\sqrt{2}\right){\alpha }_2^{m+1}}{8}},$$

where ${\alpha }_1=3+2\sqrt{2}and{\alpha }_2=3-2\sqrt{2}$.

Proof. 

By using Theorem 7 and the Binet formulas for balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}& =& 17\left({\displaystyle \frac{{\alpha }_1^m+{\alpha }_2^m}{2}}\right)+8\left({\displaystyle \frac{{\alpha }_1^m-{\alpha }_2^m}{4\sqrt{2}}}\right)\hfill \\ & =& {\displaystyle \frac{(34\sqrt{2}+8){\alpha }_1^m+(34\sqrt{2}-8){\alpha }_2^m}{4\sqrt{2}}}\hfill \\ & =& {\displaystyle \frac{\left(68+8\sqrt{2}\right){\alpha }_1^m+(68-8\sqrt{2}){\alpha }_2^m}{8}}.\hfill \end{array}$$

The other case can be proven similarly. □

The following theorem gives the relationship of consecutive odd terms for the sequence of $B_3$-Lucas subbalancing numbers with the other integer sequences.

Theorem 9.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number, $C_m$ denote the mth Lucas-balancing number, $b_m$ denote the mth cobalancing number, and $c_m$ denote the mth Lucas-cobalancing number. Then, for $m\ge 1$

$${\left(CS{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m-1}=2[30(2b_{m+1}+1)-17C_m],$$

$${\left(CS{B}_3\right)}_{2m+1}+{\left(CS{B}_3\right)}_{2m-1}=2(13c_{m+1}+17C_m).$$

Proof. 

From Theorem 7 and the relationships of balancing numbers to both Lucas-balancing and cobalancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m-1}& =& [{\left(CS{B}_3\right)}_{2m+1}+{\left(CS{B}_3\right)}_{2m}]-[{\left(CS{B}_3\right)}_{2m}+{\left(CS{B}_3\right)}_{2m-1}]\hfill \\ & =& [17(C_{m+1}+C_m)-8(B_{m+1}-B_m)]\hfill \\ \hfill & & -[(17C_m+8B_m)+(17C_m-8B_m)]\hfill \\ & =& 17[(3B_{m+1}-B_m)+(B_{m+1}-3B_m)]-8(B_{m+1}-B_m)-34C_m\hfill \\ & =& 60(B_{m+1}-B_m)-34C_m\hfill \\ & =& 60\left({\displaystyle \frac{b_{m+2}-b_{m+1}}{2}}-{\displaystyle \frac{b_{m+1}-b_m}{2}}\right)-34C_m\hfill \\ & =& 60(2b_{m+1}+1)-34C_m\hfill \\ & =& 2[30(2b_{m+1}+1)-17C_m].\hfill \end{array}$$

Similarly, from Theorem 7 and the relationship of the Lucas-cobalancing numbers to both balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m+1}+{\left(CS{B}_3\right)}_{2m-1}& =& [{\left(CS{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m}]+[{\left(CS{B}_3\right)}_{2m}+{\left(CS{B}_3\right)}_{2m-1}]\hfill \\ & =& [17(C_{m+1}-C_m)-8(B_{m+1}+B_m)]\hfill \\ \hfill & & +[(17C_m+8B_m)+(17C_m-8B_m)]\hfill \\ & =& [34c_{m+1}-8c_{m+1}]+34C_m\hfill \\ & =& 26c_{m+1}+34C_m\hfill \\ & =& 2(13c_{m+1}+17C_m).\hfill \end{array}$$

□

The following theorem gives the relationship of consecutive even terms of the sequence of $B_3$-Lucas subbalancing numbers with the other integer sequences.

Theorem 10.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number, $B_m$ denote the mth balancing number, $b_m$ denote the mth cobalancing number and $c_m$ denote the mth Lucas-cobalancing number. Then, for $m\ge 1$

$${\left(CS{B}_3\right)}_{2m}-{\left(CS{B}_3\right)}_{2m-2}=2(8B_m+13c_m),$$

$${\left(CS{B}_3\right)}_{2m}+{\left(CS{B}_3\right)}_{2m-2}=2[8B_m+30(2b_m+1)].$$

Proof. 

From Theorem 7 and the relationships Lucas-cobalancing numbers to both balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}-{\left(CS{B}_3\right)}_{2m-2}& =& [{\left(CS{B}_3\right)}_{2m}-{\left(CS{B}_3\right)}_{2m-1}]+[{\left(CS{B}_3\right)}_{2m-1}-{\left(CS{B}_3\right)}_{2m-2}]\hfill \\ & =& [(17C_m+8B_m)-(17C_m-8B_m)]\hfill \\ \hfill & & +[17(C_m-C_{m-1})-8(B_m+B_{m-1})]\hfill \\ & =& 16B_m+26c_m\hfill \\ & =& 2(8B_m+13c_m).\hfill \end{array}$$

Similarly, from Theorem 7 and the relationships of balancing numbers to both Lucas-balancing and cobalancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}+{\left(CS{B}_3\right)}_{2m-2}& =& [{\left(CS{B}_3\right)}_{2m}-{\left(CS{B}_3\right)}_{2m-1}]+[{\left(CS{B}_3\right)}_{2m-1}+{\left(CS{B}_3\right)}_{2m-2}]\hfill \\ & =& [(17C_m+8B_m)-(17C_m-8B_m)]\hfill \\ \hfill & & +[17(C_m+C_{m-1})-8(B_m-B_{m-1})]\hfill \\ & =& 16B_m+60(2b_m+1)\hfill \\ & =& 2[8B_m+30(2b_m+1)].\hfill \end{array}$$

□

Theorem 11.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and $B_m$ denote the mth balancing number. Then

$$B_{2m}\mid {\left(CS{B}_3\right)}_{2m}^2-{\left(CS{B}_3\right)}_{2m-1}^2(m\ge 1),$$

$$B_{2m+1}\mid {\left(CS{B}_3\right)}_{2m+1}^2-{\left(CS{B}_3\right)}_{2m}^2(m\ge 0).$$

Proof. 

From Theorem 7 and the relations between balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m}^2-{\left(CS{B}_3\right)}_{2m-1}^2& =& [{\left(CS{B}_3\right)}_{2m}+{\left(CS{B}_3\right)}_{2m-1}][{\left(CS{B}_3\right)}_{2m}-{\left(CS{B}_3\right)}_{2m-1}]\hfill \\ & =& [(17C_m+8B_m)+(17C_m-8B_m)][(17C_m+8B_m)-(17C_m-8B_m)]\hfill \\ & =& 544C_m{B}_m\hfill \\ & =& 272B_{2m}.\hfill \end{array}$$

Thus, we obtain

$$B_{2m}\mid {\left(CS{B}_3\right)}_{2m}^2-{\left(CS{B}_3\right)}_{2m-1}^2.$$

From Theorem 7, the relationships of balancing numbers to both Lucas-balancing and cobalancing numbers, and the relationships of Lucas-cobalancing numbers to both Lucas-balancing and balancing numbers, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2m+1}^2-{\left(CS{B}_3\right)}_{2m}^2& =& [{\left(CS{B}_3\right)}_{2m+1}+{\left(CS{B}_3\right)}_{2m}][{\left(CS{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m}]\hfill \\ & =& [17(C_{m+1}+C_m)-8(B_{m+1}-B_m)][17(C_{m+1}-C_m)-8(B_{m+1}+B_m)]\hfill \\ & =& 1560(2b_{m+1}+1)c_{m+1}\hfill \\ & =& 1560(B_{m+1}-B_m)(B_{m+1}+B_m)\hfill \\ & =& 1560B_{2m+1}.\hfill \end{array}$$

Thus, we obtain

$$B_{2m+1}\mid {\left(CS{B}_3\right)}_{2m+1}^2-{\left(CS{B}_3\right)}_{2m}^2.$$

□

Theorem 12.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and $B_n$ denote the nth balancing number. Then

$${\left(CS{B}_3\right)}_{2n+m}={\left(CS{B}_3\right)}_m{B}_{n+1}-{\left(CS{B}_3\right)}_{m-2}{B}_n$$

for any two positive integers n and m, where $m\ge 2$.

Proof. 

This theorem is proved by induction on n. It is easily seen that the assertion is true for $n=1$. Assuming the assertion is true for $n\le k$, we have

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_{2k+m+2}& =& 6{\left(CS{B}_3\right)}_{2k+m}-{\left(CS{B}_3\right)}_{2k+m-2}\hfill \\ & =& 6[{\left(CS{B}_3\right)}_m{B}_{k+1}-{\left(CS{B}_3\right)}_{m-2}{B}_k]-[{\left(CS{B}_3\right)}_m{B}_k-{\left(CS{B}_3\right)}_{m-2}{B}_{k-1}]\hfill \\ & =& {\left(CS{B}_3\right)}_m(6B_{k+1}-B_k)-{\left(CS{B}_3\right)}_{m-2}(6B_k-B_{k-1})\hfill \\ & =& {\left(CS{B}_3\right)}_m{B}_{k+2}-{\left(CS{B}_3\right)}_{m-2}{B}_{k+1}.\hfill \end{array}$$

Thus, it is shown that the assertion is true for $n=k+1$. □

We provide the following examples that will be useful in understanding more clearly the algebraic identity given in Theorem 12.

For $n=1$, since $B_1=1$ and $B_2=6$, we obtain

$${\left(CS{B}_3\right)}_{m+2}=6{\left(CS{B}_3\right)}_m-{\left(CS{B}_3\right)}_{m-2},$$

which is actually the recurrence relation of $B_3$-Lucas subbalancing numbers given in Theorem 6.

For $n=2$ and $m=2$, as $B_2=6,B_3=35$ and ${\left(CS{B}_3\right)}_0=17,{\left(CS{B}_3\right)}_2=59,{\left(CS{B}_3\right)}_6=1963$, we obtain

$${\left(CS{B}_3\right)}_6=35{\left(CS{B}_3\right)}_2-6{\left(CS{B}_3\right)}_0.$$

Theorem 13.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number. Then

$${\left(CS{B}_3\right)}_m^2={\left(CS{B}_3\right)}_{m-2}{\left(CS{B}_3\right)}_{m+2}-2248(m\ge 2).$$

Proof. 

From Theorem 3 and Corollary 6, we obtain

$$\begin{array}{ccc}\hfill {\left(CS{B}_3\right)}_m^2& =& [{\left(S{B}_3\right)}_{m+2}-3{\left(S{B}_3\right)}_m][3{\left(S{B}_3\right)}_m-{\left(S{B}_3\right)}_{m-2}]\hfill \\ & =& 3{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_m-{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_{m-2}-9{\left(S{B}_3\right)}_m^2+3{\left(S{B}_3\right)}_m{\left(S{B}_3\right)}_{m-2}\hfill \\ & =& 3{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_m-9{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_{m-2}-{\left(S{B}_3\right)}_m^2+3{\left(S{B}_3\right)}_m{\left(S{B}_3\right)}_{m-2}-2248\hfill \\ & =& [{\left(S{B}_3\right)}_m-3{\left(S{B}_3\right)}_{m-2}][3{\left(S{B}_3\right)}_{m+2}-{\left(S{B}_3\right)}_m]-2248\hfill \\ & =& {\left(CS{B}_3\right)}_{m-2}{\left(CS{B}_3\right)}_{m+2}-2248.\hfill \end{array}$$

□

Theorem 14.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and $B_m$ denote the mth balancing number. Then, for $m\ge 1$

$${\left(CS{B}_3\right)}_{m+2}{\left(CS{B}_3\right)}_{m+1}-{\left(CS{B}_3\right)}_m{\left(CS{B}_3\right)}_{m-1}=2248B_{m+1}.$$

Proof. 

By using Theorem 4 and Corollary 6, we get

$$\begin{array}{cc}\hfill {\left(CS{B}_3\right)}_{m+2}{\left(CS{B}_3\right)}_{m+1}-{\left(CS{B}_3\right)}_m{\left(CS{B}_3\right)}_{m-1}& =[3{\left(S{B}_3\right)}_{m+2}-{\left(S{B}_3\right)}_m][3{\left(S{B}_3\right)}_{m+1}-{\left(S{B}_3\right)}_{m-1}]\hfill \\ \hfill & -[{\left(S{B}_3\right)}_{m+2}-3{\left(S{B}_3\right)}_m][{\left(S{B}_3\right)}_{m+1}-3{\left(S{B}_3\right)}_{m-1}]\hfill \\ \hfill & =8[{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_{m+1}-{\left(S{B}_3\right)}_m{\left(S{B}_3\right)}_{m-1}]\hfill \\ \hfill & =2248B_{m+1}.\hfill \end{array}$$

□

In the following five theorems, several algebraic relations between $B_3$-Lucas subbalancing numbers and $B_3$-subbalancing numbers and $b_3$-Lucas subbalancing numbers are given.

Theorem 15.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number, ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number, and $B_m$ denote the mth balancing number. Then

$${\left(CS{B}_3\right)}_m{\left(S{B}_3\right)}_{m-1}+{\left(CS{B}_3\right)}_{m-1}{\left(S{B}_3\right)}_m=281B_m(m\ge 1).$$

Proof. 

From Theorem 4 and Corollary 6, we obtain

$$\begin{array}{cc}\hfill {\left(CS{B}_3\right)}_m{\left(S{B}_3\right)}_{m-1}+{\left(CS{B}_3\right)}_{m-1}{\left(S{B}_3\right)}_m& =(3{\left(S{B}_3\right)}_m-{\left(S{B}_3\right)}_{m-2}){\left(S{B}_3\right)}_{m-1}\hfill \\ \hfill & +({\left(S{B}_3\right)}_{m+1}-3{\left(S{B}_3\right)}_{m-1}){\left(S{B}_3\right)}_m\hfill \\ \hfill & ={\left(S{B}_3\right)}_{m+1}{\left(S{B}_3\right)}_m-{\left(S{B}_3\right)}_{m-1}{\left(S{B}_3\right)}_{m-2}\hfill \\ \hfill & =281B_m.\hfill \end{array}$$

□

In the following theorem, it is given that the difference between the multiplication of the mth term of the sequence of $B_3$-Lucas subbalancing numbers with the $(m+2)$th term of the sequence of $B_3$-subbalancing numbers and the multiplication of the $(m+2)$th term of the sequence of $B_3$-Lucas subbalancing numbers with the mth term of the sequence of $B_3$-subbalancing numbers is a constant.

Theorem 16.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number. Then

$${\left(CS{B}_3\right)}_m{\left(S{B}_3\right)}_{m+2}-{\left(CS{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_m=281(m\ge 0).$$

Proof. 

By using Theorem 3, Corollary 6 and the recurrence relation of $B_3$-subbalancing numbers, we obtain

$$\begin{array}{cc}\hfill {\left(CS{B}_3\right)}_m{\left(S{B}_3\right)}_{m+2}-{\left(CS{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_m& =[{\left(S{B}_3\right)}_{m+2}-3{\left(S{B}_3\right)}_m]{\left(S{B}_3\right)}_{m+2}\hfill \\ \hfill & -[3{\left(S{B}_3\right)}_{m+2}-{\left(S{B}_3\right)}_m]{\left(S{B}_3\right)}_m\hfill \\ \hfill & ={\left(S{B}_3\right)}_{m+2}^2-6{\left(S{B}_3\right)}_{m+2}{\left(S{B}_3\right)}_m+{\left(S{B}_3\right)}_m^2\hfill \\ \hfill & ={\left(S{B}_3\right)}_{m+2}^2-[{\left(S{B}_3\right)}_{m+2}+{\left(S{B}_3\right)}_{m-2}]{\left(S{B}_3\right)}_{m+2}+{\left(S{B}_3\right)}_m^2\hfill \\ \hfill & =281.\hfill \end{array}$$

□

Theorem 17.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and ${\left(S{B}_3\right)}_m$ denote the mth $B_3$-subbalancing number. Then, for $m\ge 0$

$${\left(CS{B}_3\right)}_{2m}{\left(S{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m+1}{\left(S{B}_3\right)}_{2m}=195,$$

$${\left(CS{B}_3\right)}_{2m+1}{\left(S{B}_3\right)}_{2m+2}-{\left(CS{B}_3\right)}_{2m+2}{\left(S{B}_3\right)}_{2m+1}=34.$$

Proof. 

From Theorems 2 and 7 and the relations between balancing and Lucas-balancing numbers, we obtain

$$\begin{array}{cc}\hfill {\left(CS{B}_3\right)}_{2m}{\left(S{B}_3\right)}_{2m+1}-{\left(CS{B}_3\right)}_{2m+1}{\left(S{B}_3\right)}_{2m}& =(17C_m+8B_m)(17B_{m+1}-C_{m+1})\hfill \\ \hfill & -(17C_{m+1}-8B_{m+1})(17B_m+C_m)\hfill \\ \hfill & =297[B_{m+1}{C}_m-C_{m+1}{B}_m]-34[C_{m+1}{C}_m-8B_{m+1}{B}_m]\hfill \\ \hfill & =195.\hfill \end{array}$$

The other case can be proven similarly. □

Theorem 18.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number, ${\left(CS{b}_3\right)}_m$ denote the mth $b_3$-Lucas subbalancing number, and $C_m$ denote the mth Lucas-balancing number . Then, for $m\ge 0$

$$C_m\mid {\left(CS{B}_3\right)}_{2m}-{\left(CS{b}_3\right)}_{2m},$$

$$C_{m+1}\mid {\left(CS{B}_3\right)}_{2m+1}-{\left(CS{b}_3\right)}_{2m+1}.$$

Proof. 

It can be seen by using Theorems 5 and 7. □

The relations regarding the divisibility property in Theorem 18 can be observed with the examples given below.

For $m=1$, since $C_1=3, C_2=17$ and ${\left(CS{B}_3\right)}_2=59, {\left(CS{b}_3\right)}_2=41$ and ${\left(CS{B}_3\right)}_3=241, {\left(CS{b}_3\right)}_3=139$, we get

$$\begin{array}{c}\hfill {\left(CS{B}_3\right)}_2-{\left(CS{b}_3\right)}_2=18,\end{array}$$

$$\begin{array}{c}\hfill {\left(CS{B}_3\right)}_3-{\left(CS{b}_3\right)}_3=102.\end{array}$$

Thus, it can be seen that $C_1\mid 18$ and $C_2\mid 102$.

For $m=2$, since $C_2=17, C_3=99$ and ${\left(CS{B}_3\right)}_4=337, {\left(CS{b}_3\right)}_4=235$ and ${\left(CS{B}_3\right)}_5=1403, {\left(CS{b}_3\right)}_5=809$, we get

$$\begin{array}{c}\hfill {\left(CS{B}_3\right)}_4-{\left(CS{b}_3\right)}_4=102,\end{array}$$

$$\begin{array}{c}\hfill {\left(CS{B}_3\right)}_5-{\left(CS{b}_3\right)}_5=594.\end{array}$$

Thus, it can be seen that $C_2\mid 102$ and $C_3\mid 594$.

Theorem 19.

Let ${\left(CS{B}_3\right)}_m$ denote the mth $B_3$-Lucas subbalancing number and ${\left(CS{b}_3\right)}_m$ denote the mth $b_3$-Lucas subbalancing number. Then, for $m\ge 1$

$${\left(CS{b}_3\right)}_{2m}{\left(CS{B}_3\right)}_{2m-2}+{\left(CS{b}_3\right)}_{2m+1}{\left(CS{B}_3\right)}_{2m-1}={\left(CS{b}_3\right)}_{2m-2}{\left(CS{B}_3\right)}_{2m}+{\left(CS{b}_3\right)}_{2m-1}{\left(CS{B}_3\right)}_{2m+1}.$$

Proof. 

By using Theorems 5 and 7, we obtain

$$\begin{array}{cc}\hfill {\left(CS{b}_3\right)}_{2m}{\left(CS{B}_3\right)}_{2m-2}+{\left(CS{b}_3\right)}_{2m+1}{\left(CS{B}_3\right)}_{2m-1}& =(11C_m+8B_m)(17C_{m-1}+8B_{m-1})\hfill \\ \hfill & +(11C_{m+1}-8B_{m+1})(17C_m-8B_m)\hfill \\ \hfill & =187C_m(C_{m+1}+C_{m-1})\hfill \\ \hfill & +88(C_m{B}_{m-1}-C_{m+1}{B}_m)\hfill \\ \hfill & +136B_m{C}_{m-1}-136C_m{B}_{m+1}\hfill \\ \hfill & +64B_m{B}_{m-1}+64B_m{B}_{m+1}\hfill \\ \hfill & =(11C_{m-1}+8B_{m-1})(17C_m+8B_m)\hfill \\ \hfill & +(11C_m-8B_m)(17C_{m+1}-8B_{m+1})\hfill \\ \hfill & ={\left(CS{b}_3\right)}_{2m-2}{\left(CS{B}_3\right)}_{2m}+{\left(CS{b}_3\right)}_{2m-1}{\left(CS{B}_3\right)}_{2m+1}.\hfill \end{array}$$

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4. Conclusions

In this work, we introduced a novel fourth-order linear recursive sequence that has relation to the notion of subbalancing numbers. The terms of this sequence are associated with $B_3$-subbalancing numbers in the way Lucas-balancing numbers are associated with balancing numbers. Therefore, we called this sequence the sequence of $B_3$-Lucas subbalancing numbers. In the context of this study, we obtained some of the basic algebraic properties of this sequence and provided several algebraic identities between this sequence and the other integer sequences. Using the results obtained in this study, additional algebraic relations about this sequence can be derived in subsequent research. Moreover, it can be investigated if this sequence is connected to other integer sequences not included in this study. For instance, it can be possible to obtain some relations between $B_3$-Lucas subbalancing and triangular numbers. This is because balancing numbers are the square root of square triangular numbers, and $B_3$-Lucas subbalancing numbers are related to balancing numbers. Apart from this, some relations can be established between the sequence of $B_3$-Lucas subbalancing numbers and the sequence of $b_3$-subbalancing numbers. The motivation for this is the relationship between the sequence of $B_3$-Lucas subbalancing numbers and the sequence of $b_3$-Lucas subbalancing numbers and the relationship between the sequence of $b_3$-Lucas subbalancing numbers and the sequence of $b_3$-subbalancing numbers. Besides these, by taking into account the methods used in this study, new integer sequences can be constructed and the algebraic properties of these sequences can be investigated. For this aim, the values of D can be taken as the other terms of the sequence of balancing numbers. In this way, new integer sequences can be obtained by using the method of construction of the sequence of $B_3$-Lucas subbalancing numbers. In addition to this, some algebraic properties of the obtained integer sequences can be investigated with the help of the method mentioned in the preliminaries section. Using these properties, some algebraic relations between the obtained integer sequences and the sequences of balancing, Lucas-balancing, cobalancing, and Lucas-cobalancing numbers can be examined.