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16 pages, 17598 KB

Open AccessArticle

Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b

by Merve Güney Duman

Symmetry 2025, 17(3), 364; https://doi.org/10.3390/sym17030364 - 27 Feb 2025

Viewed by 550

Let

{(P_{k})}_{k \geq 0}

be a Padovan sequence and

{(R_{k})}_{k \geq 0}

be a Perrin sequence. Let n, m, b, and k be non-negative integers, where

2 \leq b \leq 10

. [...] Read more.

Let

{(P_{k})}_{k \geq 0}

be a Padovan sequence and

{(R_{k})}_{k \geq 0}

be a Perrin sequence. Let n, m, b, and k be non-negative integers, where

2 \leq b \leq 10

. In this paper, we are devoted to delving into the equations

R_{n} = b^{d} P_{m} + R_{k}

and

R_{n} = b^{d} R_{m} + P_{k}

, where d is the number of digits of

R_{k}

P_{k}

in base b. We show that the sets of solutions are

R_{n} \in {R_{5}, R_{6}, R_{7}, R_{8}, R_{9}, R_{10}, R_{11}, R_{12}, R_{13}, R_{14}, R_{15},

R_{16}, R_{17}, R_{19}, R_{23}, R_{25}, R_{27}}

for the first equation and

R_{n} \in {R_{0}, R_{2}, R_{3}, R_{4}, R_{5}, R_{6}, R_{7}, R_{8}, R_{9}, R_{10}, R_{11}, R_{12}, R_{13}, R_{14}, R_{15}, R_{16}, R_{17}, R_{18}, R_{20}, R_{21}}

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. Full article

(This article belongs to the Section Mathematics)

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8 pages, 234 KB

Open AccessArticle

Some New Graph Interpretations of Padovan Numbers

by Mateusz Pirga, Andrzej Włoch and Iwona Włoch

Symmetry 2024, 16(11), 1493; https://doi.org/10.3390/sym16111493 - 8 Nov 2024

Viewed by 928

Abstract

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. Full article

(This article belongs to the Section Mathematics)

17 pages, 496 KB

Open AccessArticle

On the Cube Polynomials of Padovan and Lucas–Padovan Cubes

by Gwangyeon Lee and Jinsoo Kim

Symmetry 2023, 15(7), 1389; https://doi.org/10.3390/sym15071389 - 10 Jul 2023

Cited by 1 | Viewed by 1394

Abstract

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. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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14 pages, 548 KB

Open AccessArticle

On the Padovan Codes and the Padovan Cubes

by Gwangyeon Lee and Jinsoo Kim

Symmetry 2023, 15(2), 266; https://doi.org/10.3390/sym15020266 - 18 Jan 2023

Cited by 4 | Viewed by 2323

Abstract

We present a new interconnection topology called the Padovan cube. Despite their asymmetric and relatively sparse interconnections, the Padovan cubes are shown to possess attractive recurrent structures. Since they can be embedded in a subgraph of the Boolean cube and can have a Fibonacci cube as a subgraph, and since they are also a supergraph of other structures, it is possible that the Padovan cubes can be useful in fault-tolerant computing. For a graph with n vertices, we characterize the Padovan cubes. We also include the number of edges, decompositions, and embeddings, as well as the diameter of the Padovan cubes. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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9 pages, 600 KB

Open AccessArticle

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

by Renata Passos Machado Vieira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Axioms 2022, 11(11), 598; https://doi.org/10.3390/axioms11110598 - 28 Oct 2022

Cited by 6 | Viewed by 2821

Abstract

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. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Algebra and Number Theory)

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13 pages, 287 KB

Open AccessArticle

New Types of Distance Padovan Sequences via Decomposition Technique

by Andrzej Włoch, Małgorzata Wołowiec-Musiał and Urszula Bednarz

Appl. Sci. 2022, 12(18), 9163; https://doi.org/10.3390/app12189163 - 13 Sep 2022

Cited by 1 | Viewed by 1646

Abstract

In this paper, we introduce new kinds of generalized Padovan sequences and study their properties using number decomposition techniques. In particular, we consider three types of generalized Padovan sequences defined by the same recurrence equation with distinct initial conditions which follows from special number decomposition. Using the number decomposition method, we give their mutual relations and direct binomial formulas for considered sequences. Moreover, we give some combinatorial properties of these sequences and also define their matrix generators. Full article

9 pages, 282 KB

Open AccessArticle

On (2-d)-Kernels in the Tensor Product of Graphs

by Paweł Bednarz

Symmetry 2021, 13(2), 230; https://doi.org/10.3390/sym13020230 - 30 Jan 2021

Cited by 10 | Viewed by 2011

Abstract

In this paper, we study the existence, construction and number of

(2

d)

-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of

(2

d)

-kernels in these products helps us to [...] Read more.

In this paper, we study the existence, construction and number of

(2

d)

-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of

(2

d)

-kernels in these products helps us to characterize them. Among others, we show that the existence of

(2

d)

-kernels in the tensor product does not require the existence of a

(2

d)

-kernel in their factors. Moreover, we determine the number of

(2

d)

-kernels in the tensor product of certain factors using Padovan and Perrin numbers. Full article

(This article belongs to the Section Mathematics)

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