Engin Özkan

Erzincan Binali yıldırım University · Mathematics

Axioms (ISSN 2075-1680) Abstract: In this work, we define higher-order Jacobsthal–Lucas quaternions with the help of higher-order Jacobsthal–Lucas numbers. We examine some identities of higher-order Jacobsthal–Lucas quaternions. We introduce their basic definitions and properties. We give Binet’s formula, Cassini’s identity, Catalan’s identity, d’...

In this article, we find elements of the Lucas polynomials by using two matrices. We extend the study to the n-step Lucas polynomials. Then the Lucas polynomials and their relationship are generalized in the paper. Furthermore, we give relationships between the Fibonacci polynomials and the Lucas polynomials.

In this study, the Pell numbers are placed clockwise on the vertices of the polygons with a number corresponding to each vertex. Then, a relation among the numbers corresponding to a vertex is given. Furthermore, we obtain a formula which gives the mth term of the sequence formed at the kth vertex in an n-gon. The same procedure is repeated with Pe...

In this study, we present higher order Jacobsthal numbers. Then we define higher order Jacobsthal quaternions by using higher order Jacobsthal numbers. We give the concept of the norm and conjugate for these quaternions. We express and prove some propositions related to these quaternions. Also, we find the recurrence relation, the Binet formula and...

In this paper, we bring into light, study the polygonal structure of Fibonacci polynomials that are placed clockwise on these by a number corresponding to each vertex. Also, we find the relation between the numbers with such vertices. We present a relation for obtained sequence in an n-gon yielding the-th term formed at vertices. Also, we apply the...

In this paper, we introduce the hyperbolic k-Mersenne and k-Mersenne-Lucas octonions and investigate their algebraic properties. We give Binet's formula and present several interrelations and some well-known identities such as Catalan identity, d'Ocagne identity, Vajda identity, generating functions, etc. of these octo-nions in closed form. Further...

In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana’s sequence. We give a binomial direct formula for these sequences using graph methods, and a...

We compute the cardinality of a set of Galois-invariant isomorphism classes of irreducible rank two \(\overline{{\mathbb {Q}}}_\ell \)-smooth sheaves on \(X_1-S_1\), where \(X_1\) is a smooth projective absolutely irreducible curve of genus g over a finite field \({\mathbb {F}}_q\) and \(S_1\) is a reduced divisor, with pre-specified tamely ramifie...

-circulant matrices have applied in numerical computation, signal processing, coding theory, etc. In this study, our main goal is to investigate the r-circulant matrices of generalized Fermat numbers which are shown by We obtain the eigenvalues, determinants, sum identity of matrices. Also we find upper and lower bounds for the spectral norms of ge...

In this study, a different perspective was brought to Narayana sequences and one-, two-, three- and n-dimensional recurrence relations of these sequences were created. Then, some identities ranging from one to n-dimensions of these recurrences were created.

In this study, the main goal is to investigate the r-circulant matrices of k-Fermat and k-Mersenne numbers, then to find eigenvalues, determinants of these matrices, to evaluate their different norms (Spectral and Euclidean) and finally to find the right and skew-right circulant matrices.

In this study, firstly the k-Chebsyhev sequence is defined, and some terms of this sequence are given. Then, the relations between the terms of the k-Chebsyhev sequence are presented and the generating function of this sequence is obtained. In addition, the Catalan transformation of the sequence is given and the generating function of the Catalan k...

In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions and present some of their properties. Some identities are derived for these polynomials. Furthermore, we give the Binet formulas, Catalan, Cassini, d'Ocagne identity and generating and exponential generating function of these hybrinomial quaternions. 2000 Mathematics Su...

We define d−Gaussian Fibonacci polynomials and d−Gaussian Lucas polynomials. We give the matrix representations of these polynomials. By using the Riordan method, we obtain the factorizations of the Pascal matrix including the polynomials. Also, we define the infinite d−¿Gaussian Fibonacci polynomials matrix and d−Gaussian Lucas polynomials matrix...

In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on th...

In this paper, the Pell-Narayana sequence modulo m is studied. The paper outlines the definition of Pell-Narayana numbers and some of their combinatorial links with Eulerian, Catalan and Delannoy numbers and other special functions. From the definition, the Pell-Narayana orbit of a 2-generator group for a generating pair $(x, y) \in G$ is defined,...

In this study, we define d-Tribonacci polynomials. Some combinatorial properties of the d- Tribonacci polynomials with matrix representations are obtained with the help of Riordan arrays. In addition, d- Tribonacci number sequence, which is a new generalization of this number sequence, has been obtained by considering Pascal matrix. With the help o...

The aim of this study is to investigate r-circulant matrices containing Mersenne and Fermat numbers with arithmetic indices. We obtain the eigenvalues and determinants of these matrices implicitly. In addition, limits for matrix norms and spectral norms of these matrices are obtained. Thus, the results for right and skew-right circulant matrices ap...

In this study, the known real, complex, hyperbolic and dual numbers are combined to form a hybrid number system. While creating this system, a different number sequence was created by combining Narayana numbers with real, complex, hyperbolic and dual numbers in each term. Then, some properties of this obtained sequence of numbers, Binet formula, su...

In this article, we have developed a new method for coding\decoding the Jacobsthal and Jacobsthal-Lucas sequences via matrix representations. In this method, coding\decoding is done by transforming the messages into a square matrix. This process aims to not only increase the reliability of information security technology but also to provide the abi...

We define a new generalization of Gaussian Pell-Lucas polynomials. We call it d−Gaussian Pell-Lucas polynomials. Then we present the generating function and Binet formula for the polynomials. We give a matrix representation of d−Gaussian Pell-Lucas polynomials. Using the Riordan method, we obtain the factorizations of Pascal matrix involving the po...

In this paper, the Narayana sequence modulo m is studied. The paper outlines the definition of Narayana numbers and some of their combinatorial links with Eulerian, Catalan and Delannoy numbers and other special functions. From the definition, the Narayana orbit of a 2-generator group for a generating pair (x, y) ∈ G is defined, so that the lengths...

In this paper, we have introduced a new infinite cyclic group called the k-Fibonacci group and studied its algebraic properties. Further, we have obtained periods for k-step Fibonacci sequences in the 2-generator groups such as S_3 , D_3 , A_3 , Q_8 and in the 3-generator group, Q_8 × Z_2m .

In this paper, we introduce the hyperbolic \(k-\)Jacobsthal and \(k-\)Jacobsthal-Lucas quaternions. We present generating functions, Binet formula, Catalan’s identity, Vajda’s identity etc. for the hyperbolic k-Jacobsthal and \(k-\)Jacobsthal-Lucas quaternions.

We present a new family of Gauss (k, t)-Horadam numbers and obtain Binet formula of this family. We give the relationship between this family and the known Gauss t-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions of this family.

Unlike other fuzzy set theories, the theory of Linear Diophantine fuzzy set (LDFS) aims to overcome uncertainties via the reference parameters. The Parameters provide flexibility and efficiency to LDFS in handling the uncertain data. In this study, we introduce new distance measure for linear Diophantine fuzzy sets. Then, we studied the properties...

In the context of so much uncertainty with coronavirus variants and official mandate based on seemingly exaggerated predictions of gloom from epidemiologists, it is appropriate to consider a revised model of relative simplicity, because there can be dangers in developing models which endeavour to account for too many variables. Predictions and proj...

On generalized (k, r)-Pell and (k, r)-Pell-Lucas numbers Abstract: We introduce new kinds of k-Pell and k-Pell-Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k, r)-Pell and (k, r)-Pell-Lucas numbers. These sequences differ both according to the value of the natural number k and the val...

In this paper, inspiring Hosoya’s triangle, we define a new Narayana triangle. Then, we represent this Narayana triangle geometrically on the plane. In addition, we give some identities and properties of the new Narayana triangle.

This paper builds on Roettger and Williams’ extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations wit...

We introduce a new generalization of Gaussian Pell polynomials. We call it [Formula: see text]-Gaussian Pell polynomials. Then we present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representation of [Formula: see text]-Gaussian Pell polynomials. We introduce the matrix as binary representations accordi...

This paper builds on Roettger and Williams' extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations wit...

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Luca...

In this paper, we introduce the Pell-Eisenstein Series which are obtained by Pell numbers and they are a new class of Eisenstein-type series. First we see that they are well-defined and then we prove that the Pell-Eisenstein series satisfies some functional equations. Proofs are based on properties of Pell numbers and calculations.

In this paper, we define d− Gaussian Jacobsthal polynomials and d−Gaussian Jacobsthal-Lucas polynomials. We present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representations of them. We present these matrices as binary representation according to the Riordan group matrix representation. By using Riord...

We introduce the Catalan transform of the Incomplete Jacobsthal numbers. We apply the Hankel transform to the Catalan transforms of these numbers. We calculate determinants of matrixes formed with [Formula: see text]by using Hankel transform. Then we define the incomplete [Formula: see text]-Jacobsthal polynomials. Then we examine the recurrence re...

In this work, we investigate the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. We give Binet’s Formula, Cassini’s identity, Catalan’s identity, d’Ocagne identity, generating functions of the hyperbolic k-Jacobsthal and k-Jacobsthal–Lucas octonions. Also, we present many properties of these octonions.

In this work, we investigate the hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas octonions. We give Binet's Formula, Cassini's identity, Catalan's identity, d'Ocagne identity, generating functions of the hyperbolic k-Jacobsthal and k-Jacobsthal-Lucas octonions. Also, we present many properties of these octonions.

We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet's formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicom...

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas–Eisenstein Series. First, we show that the Pell-Lucas–Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas...

We give the bicomplex Gaussian Fibonacci and the bicomplex Gaussian Lucas numbers and establish the generating functions and Binet’s formulas related to these numbers. Also, we present the summation formula, matrix representation and Honsberger identity and their relationship between these numbers. Finally, we show the relationships among the bicom...

We give a generalization of Mersenne hybrid numbers. We find the Binet formula, the generating function, the sum, the character, the norm and the vector representation of the generalizition. We obtain some relations among this generalizition and well known hybrid numbers. Then we present some important identities, Cassini, Catalan, Vajda, D’ocagne,...

In this work, we bring to light the properties of newly formed polynomial sequences at each vertex of Pell polynomial sequences placed clockwise at each vertex in the n-gon. We compute the relation among the polynomials with such vertices. Moreover, in an n-gon, we generate a recurrence relation for a sequence giving the mth term formed at the kth....

We define Padovan hybrid quaternions by using Padovan hybrid numbers and Padovan quaternion. We give the basic operation properties of Padovan hybrid quaternion numbers. We give some properties and identities such as the Binet formula, sum formula, the matrix representation, characteristic equation, norm, characteristic and generating function for...

In this paper, we define biGaussian Pell and Pell-Lucas Polynomials. We give Binet‘s formulas, generating functions, Catalan’s identities, Cassini’s identities for these polynomials. Matrix presentations of biGaussian Pell and Pell-Lucas polynomials are found. Also, NegabiGaussian Pell and Pell-Lucas Polynomials are defined. Finally, we give some p...

In this study, we examined the Catalan transformations of the Gaussian Fibonacci numbers. We investigated the properties of this series. Then we gave some terms of the sequence and represented it with a matrix. We have obtained some sequences defined from the terms of the Catalan transformations of the Gaussian Fibonacci numbers. At the same time,...

In this paper, we de�fine a new family of (k; t)-Horadam numbers and obtain Binet formula for this family. We give the relationship between this family and the known generalized t-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions...

In this paper, an expansion of the classical hyperbolic functions is presented and studied. Also, many features of the [Formula: see text]-Jacobsthal hyperbolic functions are given. Finally, we introduced some graph and curved surfaces related to the [Formula: see text]-Jacobsthal hyperbolic functions.

We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Jacopsthal hybrid numbers, Jacopsthal-Lucas hybrid numbers and Mersenne hybrid numbers. Then we present some impo...

In this paper, we introduce the incomplete Vieta-Pell and Vieta-Pell-Lucas polynomials. We give some properties, the recurrence relations and the generating function of these polynomials with suggestions for further research.

The Fibonacci sequences, which have an increasing place in scientific studies and are the most well-known among the number sequences, first appeared in Leonardo Fibonacci's book 'Liber Abaci', which contains many basic problems, in a rabbit problem. Fibonacci sequences were obtained by writing down the monthly total number of new rabbits born by ma...

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Nara...

In this paper, we study a generalization of Narayana's numbers and Padovan's numbers. This generalization also includes a sequence whose elements are Fibonacci numbers repeated three times. We give combinatorial interpretations and a graph interpretation of these numbers. In addition, we examine matrix generators and determine connections with Pasc...

We define the generalized (k, r) – Gauss Pell numbers by using the definition of a distance between numbers. Then we examine their properties and give some important identities for these numbers. In addition, we present the generating functions for these numbers and the sum of the terms of the generalized (k,r)- Gauss Pell numbers.

In this study, 〖CS〗_(k,n) of S_(k,n) Catalan transformation of 𝑘−Jacobsthal-Lucas sequences is defined. S_(k,n) Catalan transformation of 𝑘−Jacobsthal-Lucas S_(k,n) sequences is obtained.In addition the transformation of CS_(k,n) is written as the product of the Catalan matrix C, which is the lower triangular matrix, and the S_k matrix of type 𝑛 𝑥...

We introduce advances in the study of k-Pell and k-Pell-Lucas numbers. Then we give some properties of these families and the generating functions of the families for some k. In addition, we identify the relationships between the family of k-Pell numbers and the classical Pell numbers as well as the family of k-Pell-Lucas numbers and the classical...

In this paper, we define a new family of Gauss 𝑘 − Lucas numbers. We give the relationships between the family of the Gauss 𝑘 − Lucas numbers and the known Gauss Lucas numbers. We also define the generalized polynomials for these numbers. We also obtain some interesting properties of the polynomials. We also give the relationships between the gener...

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we...

This paper introduces self-similarity inherent in planar Milich centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequ...

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of...

In the present paper, we propose some properties of the new family-generalized Fibonacci numbers which related to generalized Fibonacci numbers. Moreover, we give some identities involving binomial coefficients for-generalized Fibonacci numbers.

In this study, we present the Catalan transforms of the k-Pell sequence, the k-Pell-Lucas sequence and the Modified k-Pell sequence and examine the properties of the sequences. Then we apply the Hankel transform to the Catalan transforms of the k-Pell sequence, the Catalan transform of the k-Pell-Lucas sequence and the Catalan transform of the Modi...

In this paper, we define the new families of Gauss k−Fibonacci polynomials. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k− Fibonacci polynomials and the known Gauss Fibonacci polynomials. We also define the generalized polynomials for these numbers. We also obtain some interesting pr...

In this paper, we define the new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k-Jacobsthal numbers and the known Gauss Jacobsthal numbers, the family of the Gauss k-Jacobsthal-Lucas numbers and the known Gaus...

The relations of the sum and difference of two Fibonacci numbers, in which the difference of subscripts is even, was given by Koshy. In this study, the relations of the sum and difference of two Fibonacci numbers, in which is the difference of subscripts is odd, is introduced by using the generating functions. Similar results are given for Lucas nu...

Tribonacci numbers are placed on the corners of polygons clockwise with each corner receiving a term. Then, it is argued whether there is a relationship among the numbers in the same corners. It is shown to be able to find m th term corresponding to the corner Ak in an n-gon by a relation: (−1) + = (−2) + + (−3) + + (−4) + where, is a Tribonacciseq...

We introduce a new generalization of the known Jacobstal sequence and we call Jacobstal-Like sequence J k,n and then we study on Jacobstal-Like sequence J k,n and k-Jacobsthal-Lucas sequence j k,n side by side by introducing two special matrices for these sequences. After that, by using these matrices we obtain Binet formula for Jacobstal-Like sequ...

In this study, we present Catalan transform of the k−Jacobsthal sequence and examine the properties of the sequence. Then we put in for the Hankel transform to the Catalan transform of the k−Jacobsthal sequence. Furthermore, we acquire an interesting characteristic related to determinant of Hankel transform of the sequence.

Catalan Transform of The 𝒌−Lucas Numbers Abstract In this study, the 𝐶𝐿𝑘,𝑛 description of Catalan transformation of 𝑘−Lucas 𝐿𝑘,𝑛 sequences was given. The 𝐶𝐿𝑘,𝑛 generating function of Catalan transformation of 𝑘−Lucas 𝐿𝑘,𝑛 sequences was obtained. And also, 𝐶𝐿𝑘,𝑛 transformation was written as the multiplying of Catalan matris C which is the lower tri...

In this study, we define k-Fibonacci Polynomials by using terms of a new family of Fibonacci numbers given by Mikkawy and Sogabe in [5]. We compare the polynomials with known Fibonacci polynomial. Furthermore, we show the relationship between Pascal’s triangle and the coefficient of the k-Fibonacci polynomials. We give some important properties of...

We define the Gauss Fibonacci polynomials. Then we give a formula for the Gauss Fibonacci polynomials by using the Fibonacci polynomials. The Gauss Lucas polynomials are described and the relation with Lucas polynomials are explained. We show that there is a relation between the Gauss Fibonacci polynomials and the Fibonacci polynomials. The Gauss L...

In this study, we define 3-Fibonacci Polynomials by using terms of a new family of Fibonacci numbers was given by Mikkawy and Sogabe (2010). We give some important properties of the polynomial. Then, we compare the polynomials with known Fibonacci polynomial. We expressed these polynomials using the Fibonacci polynomials. Furthermore, we prove some...

n this article, we find elements of the Lucas polynomials by using two matrices. We extend the study to the n-step Lucas polynomials. Then the Lucas polynomials and their relationship are generalized in the paper. Furthermore, we give relationships between the Fibonacci polynomials and the Lucas polynomials. Keywords: Fibonacci numbers, Lucas numb...

A Sequence Bounded Above by the Lucas NumbersEngin Özkan1Ali Aydoğdu*2Aykut Göçer1AbstractIn this work, we consider the sequence whose

In this work, we consider the sequence whose term is the number of ℎ-vectors of length. The set of integer vectors is introduced. For 2, the cardinality of is the Lucas number is showed. The relation between the set of ℎ-vectors and the set of integer vectors is given.

Abstract: In the present study, we define new 2-Fibonacci polynomials by using terms of a new family of Fibonacci numbers given in [4]. We show that there is a relationship between the coefficient of the 2-Fibonacci polynomials and Pascal’s triangle. We give some identities of the 2-Fibonacci polynomials. Afterwards, we compare the polynomials with...

In this work, we prove some properties of a family of Fibonacci numbers and a family of Lucas numbers. Also, we give some identities between the family of Fibonacci numbers and the family of Lucas numbers

ABSTRACT We prove some theorems concerning a family of Fibonacci numbers defined in [11]. We give some relationship between the family of Fibonacci and Lucas numbers. Then we define a new family of k-Lucas numbers and prove some properties of it related to the Lucas numbers and the family of Lucas numbers. Keywords: recurrance relation, fibonacci n...

here has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a n...

We define a set L(n) of vectors with positive integral entries. We show that the cardinality of L(n) is the nth Lucas number Ln, for n ≥ 1. We then show that the number l(n) of M-sequences of length n is bounded by the Lucas number Ln, for n ≥ 1. This is an analogue of similar statements with Fibonacci numbers Fn instead of Lucas numbers.

We prove some theorems concerning the truncated Lucas sequences. We also state some conjectures concerning the period (mod m) of truncated Lucas sequences, where the integer m is at least 2. In addition, we present some computer-generated results that tend to support these conjectures.

In this study, we will answer the question “What can we say about the factor rings of Eisenstein integers which arise naturally when we consider the factor rings of the ring of integers which is the fundamental concept of abstract algebra. In other words, we will characterize the structure of factor rings for the ring of Eisenstein integers.

For prime numbers, we examined Catalan numbers in modula arithmetic and proved some theorems about them. Also, some theorems concerning with Catalan numbers for mod p, mod p 2 and mod p 3 such that p were given.. Katalan Sayılar ve Modüler Aritmetik ÖZET Modüler aritmetikte Katalan sayıları asal sayılar için inceledik ve onunla ilgili bazı teoremle...

We prove some theorems concerning truncated Fibonacci sequences, which are defined below in Section 2. We also state some conjectures concerning the period (mod m) of truncated Fibonacci sequences, where the integer m is greater than or equal to 2. In addition, we present some computer-generated results that tend to support these conjectures.

We have given formulas to find the term of the sequence constituted 3-step Fibonacci sequences in a nilpotent group with exponent p and nilpotency class n ( is a prime number).

In this paper we have constituted 3-step Fibonacci sequences by the three generating elements of a group of exponent p (p is prime) and nilpotency class n.

In this paper, we have constituted 3-step general Fibonacci sequences in a nilpotent group with exponent p (p is a prime number) and nilpotency class 4 and given formulas to find the α term of the sequence.

We have proved two original theorem concerning Wall number of the the 3-step Fibonacci sequences. We give five conjectures concerning 3-step Fibonacci sequence. Furthermore, we also give a computer verification of those conjectures for the primes less than 5×105.

We have proved two original theorem concerning Wall number of the the three-step Fibonacci sequences constructed by three elements of any finite p-group of exponent p and nilpotency class 4.

In this paper, we have constituted 3-step general Fibonacci sequences in a nilpotent group with exponent p (p is a prime number) and nilpotency class 4 and given formulas to find the term of the sequence.

In this work, we have proved that, for the 3-step general Fibonacci recurrence and any finite p-group of exponent p and nilpotency class 2, the length of a fundamental period of any loop satisfying the recurrence must divide the period of the ordinary 3-step general Fibonacci sequence in the field GF(p).

We have proved two original theorems concerning the periodicity and Wall number of the 2-step general Fibonacci sequences constructed by two generating elements of the modular group.