In this paper, we derive unified closed-form solutions for third-order nonhomogeneous linear recurrence relations of Leonardo-type sequences, where the input term is a polynomial. By decomposing each recurrence into homogeneous and particular components, we establish explicit for...
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Research Article10.9734/air/2026/v27i31650
Recurrence relations provide a powerful framework for understanding numerical sequences, with applications spanning both classical and modern mathematics. In earlier work, explicit iterative formulas for polynomial-type particular solutions of generalized Leonardo-type sequences...
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Research Article10.9734/ajarr/2026/v20i51361
This paper introduces two innovative third-order recurrence sequences: the generalized Jacobsthal-Narayana sequence and the co-Jacobsthal-Narayana sequence. It examines their interrelated properties, including Binet’s formulas, generating functions, Simson’s formulas, and matrix...
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Research Article10.9734/ajarr/2025/v19i5998
In this study, we define and investigate some linear third order polynomials called the generalized Horadam-Leonardo polynomials (with its two special cases, namely), (r, s)-Horadam-Leonardo and (r, s)-Horadam-Leonardo-Lucas polynomials. We give Binet’s formulas, generating funct...
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Research Article10.9734/ajarr/2023/v17i8511
It is aimed to describe generalized Bigollo sequences and to examine Bigollo and Bigollo-Lucas sequences in this article. For this purpose, we give Binet’s formulas, Simson formulas, generating functions and we introduce a lot of main features of these sequences. Also, we get som...
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Research Article10.9734/arjom/2023/v19i8690
In this paper, we define and investigate the generalized Friedrich sequences and we deal with, in detail, two special cases, namely, Friedrich and Friedrich-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these s...
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Research Article10.9734/jamcs/2023/v38i31748
In this paper, we introduce and investigate the generalized Blaise sequences and we deal with, in detail, two special cases, namely, Blaise and Blaise-Lucas sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequenc...
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Research Article10.9734/ajarr/2023/v17i1463
In this paper, we introduce and investigate the generalized Olivier sequences and we deal with, in detail, two special cases, namely, Olivier and Olivier-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequ...
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Research Article10.9734/arjom/2023/v19i1634
In this paper, closed forms of the sum formulas \(\sum{^n_k{_=}{_0}}x^kW{^2_m{_k}_+{_j}}\) for generalized p-Oresme numbers are presented. As special cases, we give sum formulas of Modified p-Oresme, p-Oresme-Lucas and p-Oresme numbers. We present the proofs to indicate how these...
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Research Article10.9734/ajarr/2022/v16i130444
In this paper, we define the binomial transform of the generalized fifth order Pell sequence and as special cases, the binomial transform of the fifth order Pell and fifth order Pell-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’...
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Research Article10.9734/ajarr/2021/v15i930423
In this study, we define the binomial transform of the generalized fourth order Pell sequence and as special cases, the binomial transform of the fourth order Pell and fourth order Pell-Lucas sequences will be introduced. We investigate their properties in details.
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Research Article10.9734/acri/2021/v21i630250
In this paper, we introduce the generalized p-Oresme sequences and we deal with, in detail, three special cases which we call them modified p-Oresme, p-Oresme-Lucas and p-Oresme sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formu...
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Research Article10.9734/ajarr/2021/v15i730410
In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how the...
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Research Article10.9734/acri/2021/v21i330235
In this paper, closed forms of the sum formulas ∑n k =0 xkWk and ∑n k=1 xkW-k for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, Hexanacci-Lucas, and other sixth-order recurrence sequences.
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Research Article10.9734/arjom/2021/v17i330285
In this paper, we investigate the recurrence properties of the generalized Hexanacci sequence under the mild assumption that the roots of the corresponding characerteristic polynomial are all distinct, and present how the generalized Hexanacci sequence at negative indices can be...
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Research Article10.9734/ajarr/2021/v15i230370
In this paper, closed forms of the summation formulas for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, Hexanacci-Lucas, sixth order Pell, sixth order Pell-Lucas, sixth order Jacobsthal, sixth order Jacobsthal-Lucas number...
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Research Article10.9734/ajarr/2020/v14i230329
In this paper, we present Binet’s formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and PentanacciLucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numb...
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Research Article10.9734/arjom/2020/v16i930224
In this paper, we introduce the generalized 6-primes sequence and we deal with, in detail, three special cases which we call them 6-primes, Lucas 6-primes and modified 6-primes sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formul...
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Research Article10.9734/acri/2020/v20i630199
In this paper, closed forms of the sum formulas for generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third-order linear recurrance sequences.
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Research Article10.9734/jsrr/2020/v26i730283
In this paper, we introduce and investigate the generalized reverse 3-primes sequences and we deal with, in detail, three special cases which we call them reverse 3-primes, reverse Lucas 3-primes and reverse modified 3-primes sequences. We present Binet’s formulas, generating fun...
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Research Article10.9734/jsrr/2020/v26i630267
In this paper, we investigate the generalized Pell-Padovan sequences and we deal with, in detail, four special cases, namely, Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell sequences. We present Binet’s formulas, generating functions, Simson form...
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Research Article10.9734/ajarr/2020/v11i230259
In this paper, closed forms of the sum formulas Σn k=0 xkW2 k ; Σn k=0 xk Σ Wk+1Wk and n k=0 xkWk+2Wk for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin numbers and the oth...
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Research Article10.9734/acri/2020/v20i430187
In this paper, closed forms of the sum formulas Σn k=0 kW3 k and Σn k=1 kW3-k for the cubes of generalized Fibonacci numbers are presented. As special cases, we give sum formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to...
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Research Article10.9734/arjom/2020/v16i630196
In this paper, we introduce the generalized 2-primes sequences and we deal with, in detail, three special cases which we call them 2-primes, Lucas 2-primes and modified 2-primes sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formu...
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Research Article10.9734/ajarr/2020/v9i230217
In this paper, we introduce the generalized Grahaml sequences and we deal with, in detail, three special cases which we call them Grahaml, Grahaml-Lucas and modified Grahaml sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas...
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Research Article10.9734/jamcs/2020/v35i230248
In this paper, closed forms of the summation formulas for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, i...
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Research Article10.9734/jamcs/2020/v35i130241
In this paper, closed forms of the summation formulas for generalized Pentanacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Pentanacci, Pentanacci-Lucas, fifth order...
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Research Article10.9734/jamcs/2019/v34i530224
In this paper, closed forms of the summation formulas for generalized Tetranacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, fourth order...
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Research Article10.9734/ajarr/2019/v7i230170
In this paper, we investigate the generalized fifth order Pell sequences and we deal with, in detail, three special cases which we call them as fifth order Pell, fifth order Pell-Lucas and modied fifth order Pell sequences.
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Research Article10.9734/arjom/2019/v15i330150
In this paper, closed forms of the summation formulas for generalized Fibonacci and Gaussian generalized Fibonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Fibonacc...
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Research Article10.9734/air/2019/v20i230154
In this paper, we investigate the generalized third order Pell sequences and we deal with, in detail, three special cases which we call them third order Pell, third order Pell-Lucas and modified third order Pell sequences. We present Binet’s formulas, generating functions, Simson...
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Research Article10.9734/ajarr/2019/v6i130144
The quaternions form a 4-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tetranacci and Tetranacci-Lucas quaternions. Furthermore, we present some properties of these quaternions and derive relationships between them. We present the generating functions, Binet...
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Research Article10.9734/arjom/2019/v15i130137
In this paper, we present linear summation formulas for generalized Hexanacci numbers and generalized Gaussian Hexanacci numbers. Also, as special cases, we give linear summation formulas of Hexanacci and Hexanacci-Lucas numbers; Gaussian Hexanacci and Gaussian Hexanacci-Lucas nu...
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Research Article10.9734/arjom/2019/v14i430136
The well known Pentanacci sequence is a fifth order recurrence sequence. In this paper, it will be dened an other generalized Pentanacci sequence and established some properties of this sequence using matrix methods.
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Research Article10.9734/arjom/2019/v14i330129
In this paper, we present linear summation formulas for generalized Pentanacci numbers and generalized Gaussian Pentanacci numbers. Also, as special cases, we give linear summation formulas of Pentanacci and Pentanacci-Lucas numbers; Gaussian Pentanacci and Gaussian Pentanacci-Lu...
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Research Article10.9734/jamcs/2019/v33i330176
In this paper, we dene Gaussian generalized Tetranacci numbers and as special cases, we investigate Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers with their properties. We present Binet's formulas, generating functions, and the summation formulas for Gaussian generali...
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Research Article10.9734/jamcs/2019/v31i330112
In this paper, we introduce the dual generalized Pandita numbers defined over the bidimensional Clifford algebra of hyperbolic numbers. As special cases, we examine the dual Pandita and dual Pandita Lucas numbers. We derive Binet formulas, construct generating functions, and esta...
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Research Article10.9734/ajarr/2026/v20i11272
In this study, we introduce and develop the concept of Dual Adrien numbers, with particular emphasis on two fundamental cases: the Dual Adrien sequence and the Dual Adrien–Lucas sequence. We conduct a systematic investigation of their structural and analytical properties, encompa...
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Research Article10.9734/ajarr/2025/v19i121229
In this study, we present a new family of number sequences called the generalized hyperbolic Pierre numbers, defined within the bidimensional Clifford algebra of hyperbolic numbers. This algebraic framework enables the extension of classical sequence theory into the hypercomplex...
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Research Article10.9734/jsrr/2025/v31i113709
This paper introduces the framework of generalized dual hyperbolic Pandita numbers, contributing a novel class of structured sequences to the expanding domain of number theory. Anchored in the principles of dual and hyperbolic systems, these constructs pave the way for exploring...
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Research Article10.9734/acri/2025/v25i81428
In this research, we introduce the generalized dual hyperbolic Edouard numbers, a novel class of number sequences that extends existing recurrence relations into a new mathematical framework. Several special cases of these numbers are examined in detail, including the dual hyperb...
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Research Article10.9734/ajarr/2025/v19i71100
In this study, we introduce the concept of Gaussian Generalized Adrien numbers, a novel extension within the framework of special number sequences. Our focus centers on two particular instances: the Gaussian Adrien numbers and the Gaussian Adrien-Lucas numbers. We systematically...
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Research Article10.9734/acri/2025/v25i71351
In this study, we introduce and investigate a new class of numerical sequences in the complex domain—Gaussian generalized Pandita numbers—which extend the classical theory of linear recurrence relations. In particular, we focus on two distinct cases: the Gaussian Pandita numbers...
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Research Article10.9734/ajarr/2025/v19i71079
This paper examines the properties and applications of Gaussian Generalized Edouard numbers, aiming to enrich the theoretical framework of number sequences. Utilizing analytical and algebraic techniques, we derive novel recurrence relations, sum formulas, and various representati...
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Research Article10.9734/ajarr/2025/v19i51028
In this work, we introduce the generalized dual hyperbolic Woodall numbers. As special cases, we study with dual hyperbolic Woodall, dual hyperbolic modified Woodall, dual hyperbolic Cullen numbers and dual hyperbolic modified Cullen numbers. Also, we present Binet’s formulas, ge...
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Research Article10.9734/acri/2024/v24i11981
In this study, we introduce the generalized hyperbolic Woodall numbers. As special cases, we study with hyperbolic Woodall, hyperbolic modified Woodall, hyperbolic Cullen numbers and hyperbolic modified Cullen numbers. We present Binet’s formulas, generating functions and the sum...
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Research Article10.9734/ajarr/2024/v18i2605
In this work, we define and investigate Gaussian generalized Woodall numbers in detail, and focus on four specific cases: Gaussian modified Woodall numbers, Gaussian modified Cullen numbers, Gaussian Woodall numbers, and Gaussian Cullen numbers. We present some identities and mat...
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Research Article10.9734/acri/2023/v23i8611
In this study, we define Gaussian generalized Guglielmo numbers in detail, and focus on four specific cases: Gaussian triangular numbers, Gaussian triangular-Lucas numbers, Gaussian oblong numbers, and Gaussian pentagonal numbers. In addition, we present some identities and matri...
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Research Article10.9734/ajarr/2023/v17i12581
In this article presents findings related to Toeplitz matrices with Mersenne numbers. We started by creating Toeplitz matrices whose components are Mersenne numbers. We then calculated the Euclidian, row and column norms of these matrices and established both lower and upper boun...
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Research Article10.9734/acri/2023/v23i7600
In this article, we present results on Toeplitz matrices with Oresme numbers components. First, the Toeplitz matrices with Oresme numbers components are created and then the Frobenius(Euclidian), row and column norms of these matrices are found. Furthermore lower and upper bounds...
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Research Article10.9734/ajarr/2023/v17i10532
In this paper, we investigate the generalized third order Jacobsthal sequences and we deal with, in detail, four special cases, namely, third order Jacobsthal, third order Jacobsthal-Lucas, modified third order Jacobsthal, third order Jacobsthal Perrin sequences. We present Binet...
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Research Article10.9734/arjom/2021/v17i230270