This paper introduces the Gaussian Lehmer sequence and presents its representation using a lower triangular Pascal matrix. Building on this matrix form, a new coding method is developed with improved capabilities for error detection and correction. The study explores the use of this coding approach in the realm of complex numbers within coding theory. Further investigation shows that the sequence follows distinct recurrence relations for odd and even terms. To unify these behaviors, a new function is proposed that captures both cases, resulting in the definition of a biperiodic sequence. This paper also provides a thorough analysis of the circulant matrix generated by the sequence, including the computation of its determinant and inverse, highlighting the promise of this matrix-based framework in advancing coding theory.

This study investigates the structural and analytical properties of harmonic and hyper-harmonic Narayana numbers, extending classical results on harmonic and hyper-harmonic numbers. We establish novel combinatorial identities and derive closed-form expressions for and ​, which represent finite sums involving reciprocals of Narayana numbers. Furthermore, we examine the spectral and Euclidean norms of circulant and r-circulant matrices generated by these numbers, together with selected special matrix forms. Through this framework, several inequalities associated with matrix norms are obtained, providing deeper insight into the interplay between Narayana-type sequences and matrix analysis. The results contribute both to the combinatorial theory of special number sequences and to the normative analysis of structured matrices.

In this paper, we study the polynomial evaluation matrix generated by the q-shifted factorial basis. We explicitly place this matrix within the broader class of generalized Vandermonde-type matrices. Since each polynomial in the underlying basis is monic and has the prescribed degree, the matrix admits a factorization into a unit lower triangular change-of-basis matrix and the classical Vandermonde matrix. Consequently, the determinant formula and the corresponding nonsingularity criterion are general basis-change consequences and are not specific to the q-shifted factorial basis. For the particular basis considered here, we determine the entries of the change-of-basis matrix explicitly in terms of q and a. Although the determinant is independent of these parameters, the factorization is not a similarity transformation; hence the characteristic polynomial, eigenvalues, singular values, and norms may still depend nontrivially on the q-structure. We specialize the standard block-inversion and Laplace-expansion techniques to obtain recursive representations for the inverse and the characteristic polynomials, including an auxiliary minor term that retains the dependence on q and a. We also identify independent sign hypotheses that are sufficient for particular Sturm conditions of the associated characteristic-polynomial sequence. Since the recurrence relation does not automatically imply the full Sturm property, the spectral conclusions are stated conditionally. More precisely, if χ _0(λ ),χ _1(λ ),… ,χ _n(λ ) satisfies the Sturm property on (0,∞ ) , together with the required endpoint sign conditions, then the eigenvalues are real, positive, simple, and the zeros of consecutive characteristic polynomials are interlaced. Fibonacci-based examples are used to illustrate the parameter-dependent algebraic and spectral behavior. When a=0 , the entire q-dependence disappears and the proposed matrix reduces exactly to the classical Vandermonde matrix for every admissible value of q.

In this research, we present a novel family of polynomials that are derived from Fermat numbers. Alongside these newly introduced polynomials, we provide a series of important identities and properties, including the Binet formula and generating functions, which are fundamental to understanding their structure. We also show that Fermat polynomials can be efficiently expressed using matrix representations, enabling an exploration of fundamental properties derived from the invariance of the determinants of these matrices. This approach provides new perspectives on the internal connections within these polynomials. In addition, many different transforms have been considered with the help of polynomials of these new number sequences, and these transforms can play a role in many areas from secure key generation to differential equation solutions. These transformations shed light on the deeper structural properties of the Fermat polynomials and reveal intricate connections between these polynomials and other well-established mathematical constructs, enhancing our understanding of their broader implications in the field of number theory and beyond.

This study explores the mathematical and computational characteristics of geometrically weighted circulant and symmetric geometric semicirculant matrices with the aim of identifying their potential as efficient structural tools in artificial intelligence (AI) architectures and data compressions. At the preliminary stage, a comprehensive mathematical framework was established, including the derivation of various matrix norms (such as spectral and Frobenius norms), determinants, and matrix inverses. The construction of these matrices is guided by Fibonacci numbers, whose intrinsic link to the golden ratio introduces a natural geometric decay pattern. This biologically inspired structure contributes to the balance, regularity, and interpretability of the resulting matrices, which are particularly well-suited for low-complexity modeling in AI systems. Subsequently, singular value decomposition (SVD) was employed to perform low-rank approximations, with a focus on evaluating information loss through Frobenius norm differences between original and reconstructed matrices. Techniques such as soft-thresholding and selective singular value removal were applied to assess data compression performance. Results demonstrated that symmetric geometric semicirculant matrices yielded smaller norm deviations, indicating superior data retention. Moreover, by tuning the geometric ratio parameter r, further improvements in matrix compactness and fidelity were achieved, especially when reducing r to values like 41. These outcomes were visually confirmed through heatmap representations, highlighting the robustness and compression potential of the proposed matrices.

In this paper, we introduce Fibonacci and Lucas quasi-quaternions by combining classical number sequences with the structure of quasi-quaternion algebra. We investigate their fundamental algebraic properties, including real and imaginary parts, conjugates, norms, and recurrence relations. We establish Binet-type formulas, generating functions, and sum formulas for these sequences in the quasi-quaternionic setting. In addition, we derive several classical identities, such as the Cassini, Catalan, d’Ocagne, Vajda, and Honsberger identities, adapted to Fibonacci and Lucas quasi-quaternions. Furthermore, we present matrix representations of these structures and obtain explicit expressions for the powers of the associated matrices. We also consider De Moivre-type formulas in the quasi-quaternion framework and analyze the behavior of these sequences under repeated operations. The graphical representations complement the theoretical results by illustrating the structural and asymptotic behavior of these quasi-quaternionic sequences.

A Ducci sequence generated by the vector A=(a1,a2,& mldr;,an)is an element of Zn is defined by (A,DA,DA2,DA3,& mldr;), where the Ducci map D:Zn -> Zn is given by DA=(|a2-a1|,|a3-a2|,& mldr;,|an-an-1|,|a1-an|). In this paper, we examine the impact of iterative Ducci transformations on Jacobsthal numbers and construct circulant and skew-circulant matrices generated by the resulting sequences. Their properties are investigated through matrix norms (Euclidean (Frobenius), spectral, and & ell;p), determinants, and eigenvalues. To extend the classical analysis, we incorporate the Convolutional Block Attention Module (CBAM) from deep learning and interpret the structured matrices as simulated image inputs. By analyzing channel-attention vectors and their variances, we assess how successive Ducci transformations influence attention distribution. The first-order transformation produces greater variance in attention weights, indicating enhanced feature discrimination, whereas higher-order transformations promote a more balanced distribution. The results highlight how Ducci transformations influence attention variance in structured matrices.

In this study, we introduce the generalized Repunit sequence and its hybrid quaternion extension derived from a parametric recurrence relation that preserves the base-10 structure of classical Repunit numbers. Fundamental properties of the proposed sequences, including the characteristic equation, generating function, and Binet-type formula, are systematically investigated. Several algebraic identities, such as bilinear index-reduction formulas, are established to demonstrate the internal structure and consistency of the construction. Numerical experiments and graphical analyses are conducted to examine the structural behavior of the generalized Repunit sequence and its hybrid quaternion counterpart. While the scalar Repunit sequence exhibits regular and predictable growth, the hybrid quaternion extension displays significantly higher structural complexity and variability. Density distributions, contour plots, histogram representations, and discrete variation measures confirm the presence of enhanced diffusion and local irregularity in the quaternion-based structure. These statistical, graphical, and numerical findings indicate that generalized Repunit hybrid quaternion sequences possess properties that are relevant to encoding, masking, and preprocessing mechanisms in applied mathematical and computational frameworks. However, this work does not propose a complete cryptographic algorithm, nor does it claim compliance with established cryptographic security standards such as NIST SP 800-22. The results should therefore be interpreted as pre-cryptographic indicators that motivate further research toward rigorous security evaluation, algorithmic development, and broader applications in areas such as coding theory, signal processing, and nonlinear dynamical systems.

In this study, we define the k-Cullen, k-Cullen–Lucas, and Modified k-Cullen sequences, and certain terms in these sequences are given. Then, we obtain the Binet formulas, generating functions, summation formulas, etc. In addition, we examine the relations among the terms of the k-Cullen, k-Cullen–Lucas, Modified k-Cullen, Cullen, Cullen–Lucas, Modified Cullen, k-Woodall, k-Woodall–Lucas, Modified k-Woodall, Woodall, Woodall–Lucas, and Modified Woodall sequences. The generating functions were derived and analyzed, especially for cases where Fibonacci numbers were assigned to parameter k. Graphical representations of the generating functions and their logarithmic transformations revealed interesting growth trends and convergence behavior. Further, by multiplying the generating functions with exponential expressions such as ek, we explored the self-similar nature and mirrored dynamics among the sequences. Specifically, it was observed that the Modified Cullen sequence exhibited a symmetric and inverse-like resemblance to the Cullen and Cullen–Lucas sequences, suggesting the presence of deeper structural dualities. Additionally, indefinite integrals of the generating functions were computed and visualized over a range of Fibonacci-indexed k values. These integral-based graphs further reinforced the phenomenon of symmetry and self-similarity, particularly in the Modified Cullen sequence. A key insight of this study is the discovery of a structural duality between the Modified Cullen and standard Cullen-type sequences, supported both algebraically and graphically. This duality suggests new avenues for analyzing generalized recursive sequences through generating function transformations. This observation provides new insight into the structural behavior of generalized Cullen-type sequences.

In order to investigate the relationship between Gaussian Fibonacci numbers and quantum numbers and to develop both a deeper theoretical understanding in this study, q-Gaussian Fibonacci, q-Gaussian Lucas quaternions and polynomials are taken with quantum integers by bringing a different perspective. Based on these definitions, the Binet formula of these number sequences is found, and some algebraic properties, important theorems, propositions and identities related to the formula are given. Thus, new perspectives are obtained in the analysis and applications of complex systems.

This paper presents a generalization of Jacobsthal and Jacobsthal Lucas numbers, introducing a broader class known as Jacobsthal and Jacobsthal Lucas p -numbers. Each instance within this class possesses a unique Binet formula, extending the classical properties of Jacobsthal numbers and Jacobsthal-Lucas to a more flexible and comprehensive framework. By deriving individual Binet formulas for each p -number, this work lays the foundation for new analytical methods that connect integer sequences, irrational proportions, and complex numbers in novel ways. These generalized formulas aim to deepen our understanding of numerical structures and open new paths for applications across coding theory, mathematical modeling, and other fields where such recursive relationships prove essential. Also, this work provides the first known closed-form Binet formulas for Jacobsthal and Jacobsthal-Lucas p-numbers, offering a novel generalization that enriches the theory of recursive integer sequences.

This paper links terraced matrices with other well-known integer sequences, such as the Hankel matrices and related Fibonacci and Lucas matrices. These, in turn, are connected with related results of Macmahon and Sloane as well as we introduce the r-Terraced matrix as a generalization of the Terraced matrix, along with its symmetric counterpart, the symmetric r-Terraced matrix. We derive key properties of these matrices, including their spectral and Euclidean norms, upper bounds for their spreads, and characteristic polynomials. To validate and exemplify the theoretical findings, we apply them to Fibonacci numbers, providing illustrative examples that strengthen the theory and confirm its accuracy. In addition to the theoretical results, we investigated how the choice of the parameter r and the matrix dimension affect the upper bounds of the spread. Our findings reveal that selecting values of r<1 and using lower-dimensional matrices lead to tighter upper bounds while reducing computational complexity. These results highlight the practical benefits of our approach, particularly in optimization-related applications where efficiency is crucial.

This paper studies the Pell-Narayana sequence modulo \(m\). It starts by defining the Pell-Narayana numbers and examining their combinatorial relationships with well-known sequences and functions, including Eulerian, Catalan, and Delannoy numbers. Building on this, the concept of a Pell-Narayana orbit is introduced for a 2-generator group with generating pair \((x, y) \in G\), which allows the analysis of the periods of these orbits. The results include explicit calculations of the Pell-Narayana periods for polyhedral and binary polyhedral groups, depending on the choice of generating pair \((x, y)\), along with a discussion of their properties. Furthermore, the paper determines the periodic lengths of Pell-Narayana orbits for the groups \(Q_8, Q_8 \times \mathbb{Z}_{2m},\) and \(Q_8 \times_\varphi \mathbb{Z}_{2m}\) for all \(m \geq 3\).

In our study we define bicomplex (p,q)- Mersenne numbers. Utilizing these numbers, we present bicomplex (p,q)- Mersenne quaternions which are a generalization of Mersenne quaternions. All these sequences have second order recurrence relations. We obtain Binet formula, the generating functions, the exponential generating function, Catalan identity, Cassini identity, D’ocagne’s identity, summation formula for both. We also introduce matrix form of bicomplex (p,q)- Mersenne quaternions. Also, we introduce a novel generalization of Mersenne numbers by incorporating Catalan numbers into the quaternionic (p,q)- Mersenne sequence. By extending Mersenne numbers to a quaternionic framework, we establish a new connection between Catalan numbers and quaternion structures. This approach provides deeper insights into the algebraic and combinatorial properties of these generalized sequences. Potential future research directions include further exploration of their structural characteristics and applications in number theory and related fields. Also, We introduced the quaternion-type Catalan transform for Mersenne numbers, demonstrating how it amplifies growth from exponential to super-exponential and faster-than-quadratic rates. This transformation enhances the sequence’s structural complexity and information density.

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 octonions in closed form. Furthermore, we investigate the relations between hyperbolic k-Mersenne octonions and hyperbolic k-Mersenne-Lucas octonions.

This paper introduces two novel sequences: the \(k-\)-division Fibonacci--Pell polynomials and the \(k-\)-division Gaussian Fibonacci--Pell polynomials. Building on the well-known Fibonacci and Pell sequences, these new sequences are defined using a division-based approach, enhancing their combinatorial and algebraic properties. We present explicit recurrence relations, generating functions, combinatorial identities, and Binet-type formulas for these sequences. A significant contribution of the study is the factorization of the Pascal matrix via the Riordan group method using the proposed polynomials. Two distinct factorizations are derived, highlighting the algebraic structure and combinatorial interpretations of the \(k-\)-division polynomials. The work not only generalizes known polynomial sequences but also provides new insights into their matrix representations and applications.

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.

In this paper, we introduced Gaussian Fermat numbers and polynomials. We provided the Binet formula, generating functions, and the exponential generating function for these numbers and polynomials. Additionally, we derived several identities for these polynomials, including the Cassini identity, Catalan identity, Vajda identity, Halton identity, numbers and polynomials can also be obtained through matrix representations and discussed key propositions based on the fact that the determinants of these matrix representations are constant. Furthermore, we explored the relationship between Gaussian Fermat numbers, polynomials, Mersenne numbers, and Jacobsthal numbers. We also presented the Catalan, Binomial, and Binomial of Catalan transformations of the Gaussian Fermat sequence and polynomials. Finally, we introduced the generating function for the Catalan transformation of the Gaussian Fermat numbers and polynomials.

In this study, we present an encoding/decoding algorithm using Fermat and Mersenne numbers. Using Fermat and Mersenne numbers we define Fermat Q-matrices and Mersenne R-matrices. We process these matrices we defined in the same way as in Fibonacci Q-matrix. We can encode any given massage with a special encryption and we can decode the encrypted massage with the method we use. We describe these process step by step. First we define coding process and then we define decoding process. In addition we give a mix model named minesweeper using Fermat Q-and Mersenne R-matrices simultaneously. The purpose of this study is not only to increase the reliability of information security technology, but also to provide the ability to verify information at a high rate.