On Third-Order Bronze Fibonacci Numbers

: In this study, we ﬁrstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and deﬁne the third-order Bronze Fibonacci, third-order Bronze Lucas and modiﬁed third-order Bronze Fibonacci sequences. Then, we deﬁne the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we ﬁnd the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.


Introduction
In the literature, the roots of the equation x 2 − x − 1 = 0 are given as where L n denotes the n-th Lucas number and F n denotes the n-th Fibonacci number. Relation (1) is the De Moivre-type identity for Fibonacci numbers [1]. Lin, in [2,3], gave the De Moivre-type identities for the tribonacci and the tetranacci numbers by using the equation Moreover, the authors in [4] obtained the De Moivre-type identities for the second-and third-order Pell numbers by using the roots of characteristic equations x 2 − 2x − 1 = 0 and x 3 − 2x 2 − x − 1 = 0, respectively. They presented a way to construct the second-order Pell and Pell-Lucas numbers and the third-order Pell and Pell-Lucas numbers. Additionally, in [5], the author studied the generalized third-order Pell numbers. In [6], the authors gave the De Moivre-type identities for the second-order and third-order Jacobsthal numbers. The second-order Bronze Fibonacci sequence or short Bronze Fibonacci sequence is given by the linear recurrence equation B n+1 = 3B n + B n−1 with initial conditions B 0 = 0 and B 1 = 1; it is also called the 3-Fibonacci Sequence and is defined as the sequence A006190 in the OEIS [7]. In [8], Kartal extended the Bronze Fibonacci numbers to the Gaussian Bronze Fibonacci numbers and obtained Binet's formula and generating functions for these numbers. In [9], the author introduced (l, 1, p + 2q, q) numbers, (l, 1, p + 2q, q) quaternions, (l, 1, p + 2q, q) symbol elements. In [10], the authors presented a special class of elements in the algebras obtained by the Cayley Dickson process, called l-elements or (l, 1, 0, 1) numbers. They gave some properties of these sequences.
It is also known that Fibonacci Numbers are used in encryption theory. In [11], a class of square Fibonacci (p + 1) × (p + 1) -matrices, which are based on the Fibonacci p numbers p = 0, 1, 2, 3, ..., with a determinant equal to ±1, was considered. The author defined a Fibonacci coding/decoding method from the Fibonacci matrices which leads to a generalization of the Cassini formula. In [12], the authors present a new method of coding/decoding algorithms using Fibonacci Q matrices. In addition to this, the authors of [13] introduced two new coding/decoding algorithms using Fibonacci Q matrices and R matrices. In [12,13], the used methods are based on the blocked message matrices. In [14], the authors present an application in cryptography and applications of some quaternion elements. In [15], the authors presented a public key cryptosystem using an Affine-Hill chipher with a generalized Fibonacci (multinacci) matrix with large power k, denoted by Q k λ , as a key.
In this paper, we give the De Moivre-type identities for the second-order Bronze Fibonacci and the third-order Bronze Fibonacci numbers derived from the characteristic equations x 2 − 3x − 1 = 0 and x 3 − 3x 2 − x − 1 = 0, respectively. Thus, we define the generalized third-order Bronze Fibonacci numbers, third-order Fibonacci numbers, thirdorder Bronze Lucas numbers and modified third-order Bronze Fibonacci numbers. We present the generating functions, Binet's formulas, Cassini's identity, matrix representation of third-order Bronze Fibonacci sequences and some interesting identities related to these sequences. Finally, we develop an encryption and decryption algorithm using an Affine-Hill chipher with the third-order Bronze Fibonacci matrix as a key. At the end of paper, we give a numerical example of an encryption and decryption algorithm.

De Moivre-Type Identity for the Second-and Third-Order Bronze Fibonacci Numbers
In this section, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci numbers. Next, we present a method for constructing the third-order Bronze Fibonacci numbers. We define the third-order Fibonacci numbers, third-order Bronze Lucas numbers, modified third-order Bronze Fibonacci numbers and generalized third-order Bronze Fibonacci numbers. We establish De Moivre-type identities for the third-order Bronze Fibonacci numbers.
The roots of the equation The De Moivre-type identity for the second-order Bronze Fibonacci numbers can be found as: where B L n represents the Bronze Lucas numbers, which form a Bronze Fibonacci sequence with the initial conditions B L 0 = 2 and B L 1 = 3, and B F n represents Bronze Fibonacci numbers with the initial conditions B F 0 = 0 and B F 1 = 1. The third-order Bronze Fibonacci numbers are related to the roots of the equation The three roots of this equation are where U = 3 2 + 4 − 64 27 , V = 3 2 − 4 − 64 27 , UV = 4 3 , and U 3 + V 3 = 4. Thus, the powers of the root α 1 can be calculated as follows: Additionally, this sequence is also called Bisection of Tribonacci Numbers in OEIS with the code A099463, [7] .

3.
{B F n } is a third-order Bronze Fibonacci sequence with the recurrence relation The sequence is also a sum of even indexed terms of Tribonacci Numbers in OEIS with the code A113300 in [7].
The first eleven terms of the above sequences are presented in the Table 1. Now, by using these three special third-order Bronze Fibonacci sequences we define a generalized third-order Bronze Fibonacci sequence as follows: The are any arbitrary numbers not all being zero, is called a generalized third-order Bronze Fibonacci sequence.
By using the sequences {B L n }, {B M n }, and {B F n }, and applying induction over n, we find Similarly, we obtain and So, we have α n 1 , α n 2 and α n 3 in terms of B L n , B M n , and B F n . Consequently, Equations (5)-(7) are called De Moivre-type identities for the third-order Bronze Fibonacci numbers.

Generating Function and Binet's Formula for the Third-Order Bronze Fibonacci Numbers
In this section, we obtain the generating functions and Binet's formulas for the thirdorder Bronze Fibonacci sequences.
Theorem 1. The generating function for the generalized third-order Bronze Fibonacci sequence {B G n } is given by where By using the recurrence relation, we find Corollary 1. The generating functions for the sequences {B L n }, {B M n } and {B F n } can be calculated as follows where where where Theorem 2. Binet's formula for the generalized third-order Bronze Fibonacci numbers is given by: Proof. We seek for constants d 1 , d 2 and d 3 such that These are found by solving the system of linear equations for n = 0, n = 1 and n = 2 Corollary 2. Binet's formulas for the sequences {B L n }, {B M n }, and {B F n } can be calculated as: 4. Some Properties of {B G n }, {B L n }, {B M n } and {B F n } In this section, we give some properties of the third-order Bronze Fibonacci sequences such as some equalities and linear sums.
Using the definitions of three third-order Bronze Fibonacci sequences, the following results can be derived easily: Theorem 3. Linear sums for the generalized third-order Bronze Fibonacci numbers are given as follows: Proof. From the linear recurrence relation of B G n+3 , we have: Summing the left and the right sides of these equations, we obtain:

By solving this equation, we obtain
In the similar way, by using the linear recurrence equation, we find: ...
and by summing side by side, we obtain then, solving this equation we obtain: Similarly, for even indexes, we have ; then, the result is obtained by solving this equation

Corollary 3.
Linear sums for the third-order Bronze Lucas sequence {B L n } are:

Cassini's Identity for the Bronze Fibonacci Numbers
In this section, we obtain the well known Cassini identity, sometimes called Simson's formulas, for the third-order Bronze Fibonacci sequences.

Theorem 4. Cassini's identity for the generalized third-order Bronze Fibonacci numbers is given by
Proof. By using the induction method, for n = 1 Let us assume that this identity is true for n = k then, by using the recurrence relation and properties of determinants, we find that (29) is satisfied for n = k + 1.

Matrix Representation of the Third-Order Bronze Fibonacci Numbers
In this section, we give the matrix representation of the the generalized third-order Bronze Fibonacci sequence. Additionally, we derive some properties of this sequence.
The Matrix representation of the generalized third-order Bronze Fibonacci sequence is given by By induction over n, we find Now, let us define a matrix B by Theorem 5. For n ≥ 4, and det B n = 1.
Proof. For n = 4, we have 115 44 34 34 13 10 10 4 3 Suppose that for n = k which proves the theorem. Similarly, by using the properties of determinants and induction over n, we find that det B n = 1.
For n ≥ 4, let us define a matrix Proof.

1.
Since BY n = Y n+1 , it can be easily shown by induction that Y n = B n−4 Y 4 . 2.
Using the definition of Y n and induction, we find Y 4 B n = B n Y 4 . 3.
From 1 and 2, it follows that Theorem 7. For n, m ≥ 4, we have Proof. From the above theorem, we have Y n+m = Y n B m , or Since the B G n+m entry is the product of the first row of the Y n and the first column of B n , the result follows. respectively.

Application: Encryption and Decryption via Third-Order Bronze Fibonacci Numbers
In this section, as a useful application of all obtained results, we give a third-order Bronze Fibonacci encryption and decryption algorithm. In this algorithm, we use the Affine-Hill chipher method for encryption using a third-order Bronze Fibonacci matrix as a key. First of all, let us list the notations which we use in the encryption and decryption algorithms: • p is the number of the characters which the sender and receiver use. We chose p to be a prime. • φ(p) is the image of the number p under the Euler Phi function. It is known that if p is prime then φ(p) = p − 1. • D is the private key of the receiver. • P 1 is any primitive root of p .
ε is a positive integer which satisfies 1 < ε < φ(p). The prime p provides a large key space for the selection of ε. This strengthens the security of the system.

Conclusions
In this paper, we define some third-order Bronze Fibonacci sequences. Additionally, we present the De Moivre-type identities for the second-and third-order Bronze Fibonacci numbers. In addition to this, we obtain the generating functions, Binet's Formulas, Cassini's identity, and matrix representation of these sequences and some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we develop a new thirdorder Bronze Fibonacci encryption and decryption algorithm in encryption theory.  Acknowledgments: The authors appreciate the anonymous referees for their careful corrections and valuable comments on the paper.

Conflicts of Interest:
The authors declare no conflict of interest.