# Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions

^{1}

^{2}

^{3}

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*Axioms*: Algebra and Number Theory)

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Combinatorial Interpretation of the Generalized Padovan Sequence

#### 2.1. A Combinatorial Model for the Padovan Sequence

**Definition**

**2.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1**.

#### 2.2. The Combinatorial Models of Tridovan, Tetradovan and Z-Dovan Sequences

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2**.

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3**.

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4**.

**Corollary**

**1.**

## 3. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Padovan, R. Dom Hans van der Laan: Modern Primitive; Architecture and Natura Press: Amsterdam, The Netherlands, 1994; pp. 1–260. [Google Scholar]
- de Spinadel, V.M.W.; A. Buitrago, R. Towards van der laan’s plastic number in the plane. J. Geom. Graph.
**2009**, 13, 163–175. [Google Scholar] - Vieira, R.P.M.; Alves, F.R.V.; Catarino, P.M.C. A historic alanalys is of the padovan sequence. Int. J. Trends Math. Educ. Res.
**2020**, 3, 8–12. [Google Scholar] [CrossRef] - Vieira, R.P.M.; Alves, F.R.V. Explorando a sequência de Padovan através de investigação histórica e abordagem epistemológica. Boletim GEPEM
**2019**, 74, 161–169. [Google Scholar] - Vieira, R.P.M. Engenharia Didática (ED): O caso da generalização e complexificação da Sequência de Padovan ou Cordonnier. Master’s Thesis, Instituto Federal de Educação, Ciência e Tecnologia do Estado do Ceará, Fortaleza, Brazil, 2020. [Google Scholar]
- Stillwell, J. Mathematics and It’s History; Springer: New York, NY, USA, 2010; pp. 1–50. [Google Scholar]
- Grimaldi, R.P. Fibonacci and Catalan Numbers; Wiley and Sons: New York, NY, USA, 2012; pp. 3–140. [Google Scholar]
- Lagrange, J.D. A combinatorial development of Fibonacci numbers in graph spectra. Linear Algebra Appl.
**2013**, 438, 4335–4347. [Google Scholar] [CrossRef] - Spreacifico, E.V.P. Novas Identidades Envolvendo os Números de Fibonacci, Lucas e Jacobsthal via Ladrilhamentos. Ph.D. Thesis, Universidade Estadual de Campinas, IMECC, São Paulo, Brazil, 2014. [Google Scholar]
- Spivey, Z.M. The Art of Proving Binomial Identities; Taylor and Francis Ltd.: London, UK, 2019; pp. 28–70. [Google Scholar]
- Benjamin, A.T.; Quinn, J.J. The Fibonacci numbers-exposed more discretely. Math. Mag.
**2003**, 76, 182–192. [Google Scholar] - Benjamin, A.T.; Quinn, J.J. Proofs That Really Count: The Art of Combinatorial Proof. Dolciani Mathematical Expositions. Math. Assoc. Am.
**2003**, 27, 1–12. [Google Scholar] - Koshy, T. Fibonacci and Lucas Numbers with Applications; Springer: New York, NY, USA, 2019; Volume 2, pp. 87–100. [Google Scholar]
- Vieira, R.P.M.; Alves, F.R.V. Propriedades das extensões da Sequência de Padovan. Rev. Eletron. Paul. CQD
**2019**, 15, 24–40. [Google Scholar] - Vieira, R.P.M.; Alves, F.R.V.; Catarino, P.M.M.C. Sequência Matricial (s1,s2,s3)-tridovan e propriedades. Rev. Eletron. Paul. CQD
**2019**, 16, 100–121. [Google Scholar] - Seenukul, P. Matrices which have a similar properties to padovan q-matrix and its generalized relations. Sakon Nakhon Rajabhat Univ. J. Sci. Technol.
**2015**, 7, 90–94. [Google Scholar] - Mehdaoui, A. Combinatoire des Transversales de la Pyramide de Pascal. Ph.D. Thesis, University of Sciences and Technology Houari Boumediene, Bab Ezzouar, Algeria, 2021. [Google Scholar]

${\mathit{P}}_{0}^{\prime}$ | ${\mathit{P}}_{1}^{\prime}$ | ${\mathit{P}}_{2}^{\prime}$ | ${\mathit{P}}_{3}^{\prime}$ | ${\mathit{P}}_{4}^{\prime}$ | ${\mathit{P}}_{5}^{\prime}$ | ${\mathit{P}}_{6}^{\prime}$ | ${\mathit{P}}_{7}^{\prime}$ | ${\mathit{P}}_{8}^{\prime}$ | ${\mathit{P}}_{9}^{\prime}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1 | 1 | 1 | 2 | 2 | 3 | 4 | 5 |

${\mathit{T}}_{0}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | ${\mathit{T}}_{5}$ | ${\mathit{T}}_{6}$ | ${\mathit{T}}_{7}$ | ${\mathit{T}}_{8}$ | ${\mathit{T}}_{9}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1 | 1 | 2 | 2 | 4 | 5 | 8 | 11 |

${\mathit{T}\mathit{e}}_{0}$ | ${\mathit{T}\mathit{e}}_{1}$ | ${\mathit{T}\mathit{e}}_{2}$ | ${\mathit{T}\mathit{e}}_{3}$ | ${\mathit{T}\mathit{e}}_{4}$ | ${\mathit{T}\mathit{e}}_{5}$ | ${\mathit{T}\mathit{e}}_{6}$ | ${\mathit{T}\mathit{e}}_{7}$ | ${\mathit{T}\mathit{e}}_{8}$ | ${\mathit{T}\mathit{e}}_{9}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 0 | 1 | 1 | 2 | 3 | 4 | 7 | 10 | 16 |

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**MDPI and ACS Style**

Vieira, R.P.M.; Alves, F.R.V.; Catarino, P.M.M.C.
Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions. *Axioms* **2022**, *11*, 598.
https://doi.org/10.3390/axioms11110598

**AMA Style**

Vieira RPM, Alves FRV, Catarino PMMC.
Combinatorial Interpretation of Numbers in the Generalized Padovan Sequence and Some of Its Extensions. *Axioms*. 2022; 11(11):598.
https://doi.org/10.3390/axioms11110598

**Chicago/Turabian Style**

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