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Journal articles on the topic 'Padovan numbers'

Author: Grafiati

Published: 4 June 2025

Last updated: 6 July 2025

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1

Korkmaz, Serdar, and Hatice Kuşak Samancı. "Some geometric properties of the Padovan vectors in Euclidean 3-space." Notes on Number Theory and Discrete Mathematics 29, no. 4 (2023): 842–60. http://dx.doi.org/10.7546/nntdm.2023.29.4.842-860.

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Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with initial conditions and general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. Up to now, various studies have been conducted on Padovan numbers and Padovan polynomial sequences. In this study, Padovan vectors are defined for the first time by using the Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm, and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in the Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

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2

Erdağ, Özgür, Serpıl Halıcı, and Ömür Deveci. "The complex-type Padovan-p sequences." Mathematica Moravica 26, no. 1 (2022): 77–88. http://dx.doi.org/10.5937/matmor2201077e.

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In this paper, we define the complex-type Padovan-p sequence and then give the relationships between the Padovan-p numbers and the complex-type Padovan-p numbers. Also, we provide a new Binet formula and a new combinatorial representation of the complex-type Padovan-p numbers by the aid of the nth power of the generating matrix of the complex-type Padovan-p sequence. In addition, we derive various properties of the complex-type Padovan-p numbers such as the permanental, determinantal and exponential representations and the finite sums by matrix methods.

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3

Goy, T. P., and S. V. Sharyn. "A note on Pell-Padovan numbers and their connection with Fibonacci numbers." Carpathian Mathematical Publications 12, no. 2 (2020): 280–88. http://dx.doi.org/10.15330/cmp.12.2.280-288.

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In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

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4

Deveci, Ömür. "On the connections among Fibonacci, Pell, Jacobsthal and Padovan numbers." Notes on Number Theory and Discrete Mathematics 27, no. 2 (2021): 111–28. http://dx.doi.org/10.7546/nntdm.2021.27.2.111-128.

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In this paper, we define the Fibonacci–Jacobsthal, Padovan–Fibonacci, Pell–Fibonacci, Pell–Jacobsthal, Padovan–Pell and Padovan–Jacobsthal sequences which are directly related with the Fibonacci, Jacobsthal, Pell and Padovan numbers and give their structural properties by matrix methods. Then we obtain new relationships between Fibonacci, Jacobsthal, Pell and Padovan numbers.

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5

R. Sivaraman, B. Ramesh,. "Generalized Padovan Sequences and Figurate Numbers." Advances in Nonlinear Variational Inequalities 27, no. 1 (2024): 308–12. http://dx.doi.org/10.52783/anvi.v27.438.

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Generalized Padovan Sequences are more general class of sequences which are viewed as general form of the classical Padovan sequences named after English mathematician Richard Padovan. Figurate numbers are special class of numbers which occur in various problems in number theory. In this paper, we discuss the limiting ratio of the recurrence relation formed by generalizing Padovan sequence and considering Figurate numbers as coefficients in two possible cases. We find very interesting numbers turning out to be limiting ratios under our assumptions.

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6

Kartal, Meral Yaşar. "Gaussian Padovan and Gaussian Perrin numbers and properties of them." Asian-European Journal of Mathematics 12, no. 06 (2019): 2040014. http://dx.doi.org/10.1142/s1793557120400148.

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In this paper, the Gaussian Padovan and Gaussian Perrin numbers are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian Padovan and Gaussian Perrin numbers are obtained by using the recurrence relation satisfied by them. Then two relations are given between Gaussian Padovan and Gaussian Perrin numbers.

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7

Lee, Gwangyeon, and Jinsoo Kim. "On the Cube Polynomials of Padovan and Lucas–Padovan Cubes." Symmetry 15, no. 7 (2023): 1389. http://dx.doi.org/10.3390/sym15071389.

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The hypercube is one of the best models for the network topology of a distributed system. Recently, Padovan cubes and Lucas–Padovan cubes have been introduced as new interconnection topologies. Despite their asymmetric and relatively sparse interconnections, the Padovan and Lucas–Padovan cubes are shown to possess attractive recurrent structures. In this paper, we determine the cube polynomial of Padovan cubes and Lucas–Padovan cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular, they can be expressed with convolved Padovan numbers and Lucas–Padovan numbers. In particular, the coefficients of the cube polynomials represent the number of hypercubes, a symmetry inherent in Padovan and Lucas–Padovan cubes. Therefore, cube polynomials are very important for characterizing these cubes.

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8

KARAASLAN, Nusret. "On Gaussian Jacobsthal-Padovan Numbers." Cumhuriyet Science Journal 43, no. 2 (2022): 277–82. http://dx.doi.org/10.17776/csj.1025269.

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Gaussian Jacobsthal-Padovan numbers have been the central focus of this paper and firstly this number sequence has defined. Later, we have given the proof of the generating function of the Gaussian Jacobsthal-Padovan sequence. After that by using generating function, we have given the proof of the Binet formula for this number sequence. Additionally, we have investigated some properties such as Simson identity, summation formulas of this sequence. Finally, we have obtained some matrices whose elements are Gaussian Jacobsthal-Padovan numbers.

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9

Pirga, Mateusz, Andrzej Włoch, and Iwona Włoch. "Some New Graph Interpretations of Padovan Numbers." Symmetry 16, no. 11 (2024): 1493. http://dx.doi.org/10.3390/sym16111493.

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Padovan numbers and Perrin numbers belong to the family of numbers of the Fibonacci type and they are well described in the literature. In this paper, by studying independent (1,2)-dominating sets in paths and cycles, we obtain new binomial formulas for Padovan and Perrin numbers. As a consequence of graph interpretation, we propose a new dependence between Padovan and Perrin numbers. By studying special independent (1,2)-dominating sets in a composition of two graphs, we define Padovan polynomials of graphs. By the fact that every independent (1,2)-dominating set includes the set of leaves as a subset, in some cases a symmetric structure of the independent (1,2)-dominating set can be used.

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10

Yilmaz, Nazmiye, and Necati Taskara. "Matrix Sequences in terms of Padovan and Perrin Numbers." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/941673.

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The first main idea of this paper is to develop thematrix sequencesthat represent Padovan and Perrin numbers. Then, by taking into account matrix properties of these new matrix sequences, some behaviours of Padovan and Perrin numbers will be investigated. Moreover, some important relationships between Padovan and Perrin matrix sequences will be presented.

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11

Taher, Hunar Sherzad, and Saroj Kumar Dash. "On k-GENERALIZED PADOVAN NUMBERS WHICH ARE REPDIGITS IN BASE n." JP Journal of Algebra, Number Theory and Applications 64, no. 4 (2025): 395–416. https://doi.org/10.17654/0972555525021.

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Let . Then the -Padovan sequence is a generalization of the Padovan sequence. The sequence's first terms are . This paper identifies all repdigits that can be expressed as -Padovan numbers in base , where , through the application of the theory of linear forms in logarithms of algebraic numbers and a modified version of the Baker-Davenport reduction method.

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12

Duman, Merve, Refik Keskin, and Leman Hocaoğlu. "Padovan Numbers as Sum of Two Repdigits." Proceedings of the Bulgarian Academy of Sciences 76, no. 9 (2023): 1326–34. http://dx.doi.org/10.7546/crabs.2023.09.02.

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Padovan sequence $$(P_{n})$$ is given by $$P_{n}=P_{n-2}+P_{n-3}$$ for $$n\geq3$$ with initial condition $$(P_{0},P_{1},P_{2})=(1,1,1)$$. A positive integer is called a repdigit if all of its digits are equal. In this study, we examine the terms of the Padovan sequence, which are the sum of two repdigits. It is shown that the largest term of the Padovan sequence which can be written as a sum of two repdigits is $$P_{18}=114=111+3.$$

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13

Adédji, Kouèssi Norbert, Japhet Odjoumani, and Alain Togbé. "Padovan and Perrin numbers as products of two generalized Lucas numbers." Archivum Mathematicum, no. 4 (2023): 315–37. http://dx.doi.org/10.5817/am2023-4-315.

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14

Dışkaya, Orhan, and Hamza Menken. "On the bi-periodic Padovan sequences." Mathematica Moravica 27, no. 2 (2023): 115–26. http://dx.doi.org/10.5937/matmor2302115d.

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In this study, we define a new generalization of the Padovan numbers, which shall also be called the bi-periodic Padovan sequence. Also, we consider a generalized bi-periodic Padovan matrix sequence. Finally, we investigate the Binet formulas, generating functions, series and partial sum formulas for these sequences.

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15

Ddamulira, Mahadi. "Padovan numbers that are concatenations of two distinct repdigits." Mathematica Slovaca 71, no. 2 (2021): 275–84. http://dx.doi.org/10.1515/ms-2017-0467.

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Abstract Let (Pn ) n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = 1 = P 2, and P n+3 = P n+1 +Pn for all n ≥ 0. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits.

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16

Tatong, Mongkol. "Some Matrices with Padovan Q-matrix and the Generalized Relations." Progress in Applied Science and Technology 14, no. 1 (2024): 82–86. http://dx.doi.org/10.60101/past.2024.252531.

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In this paper, we establish a new -matrix for Padovan numbers and the multiplies between the -matrix and the -matrix. Moreover, we investigate the of , the of multiply the -matrix, and the of multiply the -matrix. Finally, we use these matrices to obtain elementary identities for Padovan, Perrin, and relations between numbers.

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17

Ddamulira, Mahadi. "Repdigits as sums of three Padovan numbers." Boletín de la Sociedad Matemática Mexicana 26, no. 2 (2019): 247–61. http://dx.doi.org/10.1007/s40590-019-00269-9.

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AbstractLet $$ \{P_{n}\}_{n\ge 0} $${Pn}n≥0 be the sequence of Padovan numbers defined by $$ P_0=0 $$P0=0, $$ P_1 =1=P_2$$P1=1=P2, and $$ P_{n+3}= P_{n+1} +P_n$$Pn+3=Pn+1+Pn for all $$ n\ge 0 $$n≥0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Padovan numbers.

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18

Bellaouar, Djamel, Özen Özer, and Noureddine Azzouza. "Padovan and Perrin numbers of the form $7^{t}-5^{z}-3^{y}-2^{x}$." Notes on Number Theory and Discrete Mathematics 31, no. 1 (2025): 191–200. https://doi.org/10.7546/nntdm.2025.31.1.191-200.

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Consider the Padovan sequence $\left( p_{n}\right) _{{n\geq 0}}$ given by $p_{n+3}=p_{n+1}+p_{n}$ with $p_{0}=p_{1}=p_{2}=1$. Its companion sequence, the Perrin sequence $\left( \wp _{n}\right) _{{n\geq 0}}$, follows the same recursive formula as the Padovan numbers, but with different initial values: $p_{0}=3$, $p_{1}=0$ and $p_{2}=2$. In this paper, we leverage Baker's theory concerning nonzero linear forms in logarithms of algebraic numbers along with a reduction procedure that employs the theory of continued fractions. This enables us to explicitly identify all Padovan and Perrin numbers that conform to the representation $7^{t}-5^{z}-3^{y}-2^{x}$, where $x,y,z$ and $t$ are positive integers with $0\leq x,y,z\leq t$.

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19

Vieira, Renata Passos Machado, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions." Axioms 11, no. 11 (2022): 598. http://dx.doi.org/10.3390/axioms11110598.

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There is ongoing research into combinatorial methods and approaches for linear and recurrent sequences. Using the notion of a board defined for the Fibonacci sequence, this work introduces the Padovan sequence combinatorial approach. Thus, mathematical theorems are introduced that refer to the study of the Padovan combinatorial model and some of its extensions, namely Tridovan, Tetradovan and its generalization (Z-dovan). Finally, we obtained a generalization of the Padovan combinatorial model, which was the main result of this research.

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20

İpek, Güzel, Ömür Deveci, and Anthony G. Shannon. "On the Padovan p-circulant numbers." Notes on Number Theory and Discrete Mathematics 26, no. 3 (2020): 224–33. http://dx.doi.org/10.7546/nntdm.2020.26.3.224-233.

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21

Passos Machado Vieira, Renata, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "sequence of the hyperbolic k-Padovan quaternions." Malaya Journal of Matematik 11, no. 03 (2023): 324–31. http://dx.doi.org/10.26637/mjm1103/009.

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This work introduces the hyperbolic k-Padovan quaternion sequence, performing the process of complexification of linear and recurrent sequences, more specifically of the generalized Padovan sequence. In this sense, there is the study of some properties around this sequence, deepening the investigative mathematical study of these numbers.

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22

Ddamulira, Mahadi. "On the problem of Pillai with Padovan numbers and powers of 3." Studia Scientiarum Mathematicarum Hungarica 56, no. 3 (2019): 364–79. http://dx.doi.org/10.1556/012.2019.56.3.1435.

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Abstract Let {P n}n≥0 be the sequence of Padovan numbers defined by P0 = 0, P1 = 1, P2 = 1, and Pn+3 = Pn+1 + Pn for all n ≥ 0. In this paper, we find all integers c admitting at least two representations as a difference between a Padovan number and a power of 3.

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23

Kafle, Bir, Salah Eddine Rihane, and Alain Togbé. "A note on Mersenne Padovan and Perrin numbers." Notes on Number Theory and Discrete Mathematics 27, no. 1 (2021): 161–70. http://dx.doi.org/10.7546/nntdm.2021.27.1.161-170.

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24

Vieira, Renata Passos Machado, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "Padovan sequence generalization – a study of matrix and generating function." Notes on Number Theory and Discrete Mathematics 26, no. 4 (2020): 154–63. http://dx.doi.org/10.7546/nntdm.2020.26.4.154-163.

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The Padovan sequence is a sequence similar to the Fibonacci sequence, the former being third order and the latter second. Having several applications in architecture, these numbers are directly related to plastic numbers. In this paper, the Padovan sequence is studied and investigated from the standpoint of linear algebra. With this, we will study the matrix and the generating function of the extensions of this sequence (Tridovan and Tetradovan), thus determining the generalization of this sequence.

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25

Behera, Mitashree, and Prasanta Kumar Ray. "Multiplicative Dependent Pairs in the Sequence of Padovan Numbers." Mathematica Slovaca 73, no. 5 (2023): 1135–44. http://dx.doi.org/10.1515/ms-2023-0083.

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ABSTRACT The Padovan sequence {Pn } n≥0 is a ternary recurrent sequence defined recursively by the relation Pn = P n–2 + P n–3 with initials P 0 = P 1 = P 2 = 1. In this note, we search all pairs of multiplicative dependent vectors whose coordinates are Padovan numbers. For this purpose, we apply Matveev’s theorem to find the lower bounds of the non-zero linear forms in logarithms. Techniques involving the LLL algorithm and the theory of continued fraction are utilized to reduce the bounds.

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26

Soykan, Yuksel, Vedat Irge, and Erkan Tasdemir. "A Comprehensive Study of k-Circulant Matrices Derived from Generalized Padovan Numbers." Asian Journal of Probability and Statistics 26, no. 12 (2024): 152–70. https://doi.org/10.9734/ajpas/2024/v26i12691.

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This paper presents a review of the k-circulant matrices and the generalized Padovan numbers, it further outlines the importance of these numbers and matrices with regard to matrices analysis and number theory. Considering the potential practical applications of k-circulant matrices in combination and numerical analysis, we derive explicit formulas for sum of entries, maximum column and row sum norms, Euclidean norm, spectral norm, eigenvalues and determinant of these matrices. Our research also shows the analytical relationships which exist between the usual structure of k-circulant matrices and generalized Padovan numbers that could be useful for the theoretical and practical researches.

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27

Güney Duman, Merve. "Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b." Symmetry 17, no. 3 (2025): 364. https://doi.org/10.3390/sym17030364.

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Let (Pk)k≥0 be a Padovan sequence and (Rk)k≥0 be a Perrin sequence. Let n, m, b, and k be non-negative integers, where 2≤b≤10. In this paper, we are devoted to delving into the equations Rn=bdPm+Rk and Rn=bdRm+Pk, where d is the number of digits of Rk or Pk in base b. We show that the sets of solutions are Rn∈{R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R19,R23,R25,R27} for the first equation and Rn∈{R0,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R18, R20,R21} for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in logarithms, continued fractions, and the properties of Padovan and Perrin sequences in base b. We investigate both the deep structural symmetries and the complex structures that connect recurrence relations and logarithmic forms within Diophantine equations involving special number sequences.

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28

Rihane, Salah Eddine, and Alain Togbé. "Repdigits as products of consecutive Padovan or Perrin numbers." Arabian Journal of Mathematics 10, no. 2 (2021): 469–80. http://dx.doi.org/10.1007/s40065-021-00317-1.

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AbstractA repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number of the form $$a(10^m-1)/9$$ a ( 10 m - 1 ) / 9 , for some $$m\ge 1$$ m ≥ 1 and $$1 \le a \le 9$$ 1 ≤ a ≤ 9 . Let $$\left( P_n\right) _{n\ge 0}$$ P n n ≥ 0 and $$\left( E_n\right) _{n\ge 0}$$ E n n ≥ 0 be the sequence of Padovan and Perrin numbers, respectively. This paper deals with repdigits that can be written as the products of consecutive Padovan or/and Perrin numbers.

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29

Adédji, Kouèssi N., Virgile Dossou-yovo, Salah E. Rihane, and Alain Togbé. "Padovan or Perrin numbers that are concatenations of two distinct base b repdigits." Mathematica Slovaca 73, no. 1 (2023): 49–64. http://dx.doi.org/10.1515/ms-2023-0006.

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Abstract Let {P n } n⩾0 be the Padovan sequence with initial conditions P 0=0, P 1=1, and P 2=1 and the recurrence relation P n+3=P n+1 + P n . Its companion sequence is known as the Perrin sequence {E n } n⩾0 that satisfies the same above recurrence relation with the initial conditions E 0=3, E 1=0 and E 2=2. In this paper, we determine all Padovan and Perrin numbers that are concatenations of two distinct base b repdigits with 2 ⩽ b ⩽ 9. As corollary, we prove that the largest Padovan and Perrin numbers which can be representable as a concatenations of two distinct base b repdigits are P 26 = 816 = 2244 ‾ 7 $ P_{26}=816=\overline{2244}_7 $ and E 24 = 853 = 31111 ‾ 4 $ E_{24}=853=\overline{31111}_4 $ .

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30

Falcon, Sergio. "On a Generalization of the Padovan Numbers." Journal of Advances in Mathematics and Computer Science 39, no. 3 (2024): 54–64. http://dx.doi.org/10.9734/jamcs/2024/v39i31876.

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This paper studies an extension of the classical Padovan sequence and that contains this as a particular case. Some very interesting formulas are found for the sum of these new sequences, for the sum of their squares as well as their self-convolution.

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31

Mangueira, Milena Carolina dos Santos, Renata Passos Machado Vieira, Francisco Régis Vieira Alves, and Paula Maria Machado Cruz Catarino. "The Hybrid Numbers of Padovan and Some Identities." Annales Mathematicae Silesianae 34, no. 2 (2020): 256–67. http://dx.doi.org/10.2478/amsil-2020-0019.

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AbstractIn this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation ih = −hi = ɛ + i. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.

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32

DİŞKAYA, ORHAN, and HAMZA MENKEN. "ON THE WEIGHTED PADOVAN AND PERRIN SUMS." Journal of Science and Arts 24, no. 4 (2023): 817–26. http://dx.doi.org/10.46939/j.sci.arts-23.4-a01.

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33

Došlić, Tomislav, and Luka Podrug. "Tilings of a honeycomb strip and higher order Fibonacci numbers." Contributions to Discrete Mathematics 19, no. 2 (2024): 56–81. http://dx.doi.org/10.55016/ojs/cdm.v19i2.75062.

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In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow sequence and provide combinatorial proofs for several known identities about those numbers.

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DIŞKAYA, Orhan, and Hamza MENKEN. "On the Richard and Raoul numbers." Journal of New Results in Science 11, no. 3 (2022): 256–64. http://dx.doi.org/10.54187/jnrs.1201184.

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In this study, we define and examine the Richard and Raoul sequences and we deal with, in detail, two special cases, namely, Richard and Raoul sequences. We indicate that there are close relations between Richard and Raoul numbers and Padovan and Perrin numbers. Moreover, we present the Binet-like formulas, generating functions, summation formulas, and some identities for these sequences.

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35

García Lomeli, Ana María, and Santos Hernández Hernández. "Pillai's problem with Padovan numbers and powers of two." Revista Colombiana de Matemáticas 53, no. 1 (2019): 1–14. http://dx.doi.org/10.15446/recolma.v53n1.81034.

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Let (Pn)n≥0 be the Padovan sequence given by P0 = 0, P1 = P2 = 1 and the recurrence formula Pn+3 = Pn+1 + Pn for all n ≥ 0. In this note we study and completely solve the Diophantine equation Pn - 2m = Pn1 - 2m1 in non-negative integers (n, m, n1, m1).

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36

Vieira, Renata Passos Machado, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "The Construction of Definitions Through Padovan’s Combinatorial Model: An Investigation With Didactic Engineering in an Initial Education Course for Mathematics Teachers." Acta Scientiae 25, no. 6 (2023): 366–95. http://dx.doi.org/10.17648/acta.scientiae.7728.

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Background: Given the oversight of specific topics in the history of mathematics books, this research was motivated by the extensive coverage of the Fibonacci sequence. In addition, the existence of the Padovan sequence, which is considered a Fibonacci cousin, stands out. Objectives: Investigating the Padovan sequence, building its definition through a combinatorial model using manipulative materials based on the theory of didactical situations in the initial mathematics teacher education course context. Design: The research methodological design follows didactic engineering, and a didactic teaching situation is created to investigate the Padovan sequence. Thus, manipulative material was developed to facilitate the teaching process and the construction of the sequence definition. Settings and participants: The interventions were done in the mathematics degree course of a higher education institution in Fortaleza. Five students enrolled in the component of History of Mathematics participated in the study. Data collection and analysis: Data were collected during classes, recorded through photos and audio recordings, based on didactic engineering and the theory of didactical situations. Results: The most relevant result of this study is constructing the definition of the Padovan sequence through the manipulative material. Conclusions: We concluded that the research allowed an investigation of the Padovan sequence, enabling the visualisation of its terms and their integration with other mathematical contents, contributing to teaching these numbers.

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37

Yüksel, Soykan. "A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers." Journal of Progressive Research in Mathematics 16, no. 2 (2020): 2932–41. https://doi.org/10.5281/zenodo.3973991.

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In this paper, closed forms of the sum formulas for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of the squares of Tribonacci, Tribonacci Lucas, Padovan, Perrin, Narayana and some other third order linear recurrence sequences.

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38

Mansour, Toufik, and Vincent Vajnovszki. "Restricted 123-avoiding Baxter permutations and the Padovan numbers." Discrete Applied Mathematics 155, no. 11 (2007): 1430–40. http://dx.doi.org/10.1016/j.dam.2007.03.002.

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RIHANE, Salah Eddine, Mohand Ouamar HERNANE, and Alain TOGBÉ. "The X -coordinates of Pell equations and Padovan numbers." TURKISH JOURNAL OF MATHEMATICS 43, no. 1 (2019): 207–23. http://dx.doi.org/10.3906/mat-1808-2.

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Tasyurdu, Yasemin. "Generalized Fibonacci numbers with five parameters." Thermal Science 26, Spec. issue 2 (2022): 495–505. http://dx.doi.org/10.2298/tsci22s2495t.

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In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Pell, Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, Modified k-Pell, k-Jacobsthal numbers and Fibonacci p-numbers, distance Fibonacci numbers, (2, k)-distance Fibonacci numbers, generalized (k, r)-Fibonacci numbers in the distance sense by extending the definition of a distance in the recurrence relation with two parameters and adding three parameters in the definition of this distance, simultaneously. Tiling and combinatorial interpretations of generalized Fibonacci numbers are presented, and explicit formulas that allow us to calculate the nth number are given. Also generating functions and some identities for these numbers are obtained.

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Vieira, Renata Passos Machado, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "A didactic engineering for the study of the Padovan’s combinatory model." Pedagogical Research 9, no. 3 (2024): em0206. http://dx.doi.org/10.29333/pr/14441.

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Considering the content of history of mathematics textbooks, it’s evident that their emphasis is primarily on the illustrative aspects of recurring numerical sequences, with a particular focus on the Fibonacci sequence. Unfortunately, this limited approach results in the neglect of other sequences akin to the Fibonacci numbers, thus rendering the subject challenging for teaching purposes. This study aims to address this gap by offering a concise exploration of the combinatorial aspects of the Padovan numbers, specifically through the concept of a board as initially examined by mathematicians. In line with the research methodology of didactic engineering and the teaching theory of the theory of didactic situations, two problem situations have been developed, centered on the Padovan combinatorial model, thereby contributing to the enrichment of mathematical education within initial teacher training programs. Within this framework, various strategies are introduced that rely on visualization and counting, with the objective of illustrating specific mathematical identities suitable for potential classroom applications.

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Vieira, Renata Passos Machado, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "A Historical Analysis of The Padovan Sequence." International Journal of Trends in Mathematics Education Research 3, no. 1 (2020): 8–12. http://dx.doi.org/10.33122/ijtmer.v3i1.166.

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In the present text, we present some possibilities to formalize the mathematical content and a historical context, referring to a numerical sequence of linear and recurrent form, known as Sequence of Padovan or Cordonnier. Throughout the text some definitions are discussed, the matrix approach and the relation of this sequence with the plastic number. The explicit exploration of the possible paths used to formalize the explored mathematical subject, comes with an epistemological character, still conserving the exploratory intention of these numbers and always taking care of the mathematical rigor approached

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Bilgici, Goksal. "Generalized Order–k Pell–Padovan–Like Numbers by Matrix Methods." Pure and Applied Mathematics Journal 2, no. 6 (2013): 174. http://dx.doi.org/10.11648/j.pamj.20130206.11.

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Yazlik, Yasin, D. Turgut Tollu, and Necati Taskara. "On the Solutions of Difference Equation Systems with Padovan Numbers." Applied Mathematics 04, no. 12 (2013): 15–20. http://dx.doi.org/10.4236/am.2013.412a002.

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Lee, Gwangyeon. "ON THE GENERALIZED BINET FORMULAS OF THE k-PADOVAN NUMBERS." Far East Journal of Mathematical Sciences (FJMS) 99, no. 10 (2016): 1487–504. http://dx.doi.org/10.17654/ms099101487.

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Jiang, Xiaoyu, and Kicheon Hong. "Exact Determinants of Some Special Circulant Matrices Involving Four Kinds of Famous Numbers." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/273680.

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Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively.

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Mustafa Zangana, Diyar, and Ahmet Öteleş. "Padovan Numbers by the Permanents of a Certain Complex Pentadiagonal Matrix." Journal of Garmian University 5, no. 2 (2018): 330–38. http://dx.doi.org/10.24271/garmian.346.

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Jiang, Zhaolin, Nuo Shen, and Juan Li. "Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers." Journal of Applied Mathematics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/585438.

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The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

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Taher, Hunar Sherzad, and Saroj Kumar Dash. "Repdigits base $ \eta $ as sum or product of Perrin and Padovan numbers." AIMS Mathematics 9, no. 8 (2024): 20173–92. http://dx.doi.org/10.3934/math.2024983.

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<abstract><p>Let $ \left\{E_{n}\right\}_{n\geq0} $ and $ \left\{P_{n}\right\}_{n\geq0} $ be sequences of Perrin and Padovan numbers, respectively. We have found all repdigits that can be written as the sum or product of $ E_{n} $ and $ P_{m} $ in the base $ \eta $, where $ 2\leq\eta\leq10 $ and $ m\leq n $. In addition, we have applied the theory of linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method in Diophantine approximation approaches.</p></abstract>

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BRAVO, ERIC, and NURETTİN IRMAK. "The 2-adic valuation of shifted Padovan and Perrin numbers and applications." Turkish Journal of Mathematics 48, no. 6 (2024): 1183–96. http://dx.doi.org/10.55730/1300-0098.3568.

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