Padovan Sequences Research Articles

Generalized Padovan Sequences and Figurate Numbers

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.

Recurrence and linear sequences: a three-dimensional visualization with support of GeoGebra software

"This article is an excerpt from a master's thesis developed in Brazil, in which we approach recurrent and linear sequences, given some intriguing particularities in their definitions and the scarcity of discussion of this topic in the literature of the History of Mathematics, especially with regard to its geometric representation. Thus, we aim to present the identities of Fibonacci, Lucas, Jacobsthal and Padovan in a three-dimensional visualization with the contribution of GeoGebra software. The research methodology chosen was bibliographical, exploratory in nature, where we have theoretical support in works such as Oliveira and Alves (2019), Silva (2017), Souza and Alves (2018), Vieira and Alves (2020). This research brings as results a set of geometric constructions of the identities of the proposed sequences, in three-dimensional perspective, being a support for future works developed around this theme. GeoGebra was essential in the process of constructing and visualizing the sequences, as it provided strategies for understanding the recurrence relations and the properties of the Fibonacci, Lucas, Jacobsthal and Padovan sequences, through the behavior of the visual representations of these identities."

On the Quinary Fibonacci-Padovan Sequences

In this paper, we consider the Fibonacci and Padovan sequences. We introduce the quinary Fibonacci-Padovan sequences whose compounds are the Fibonacci and Padovan sequences. We derive the Binet-like formulas, the generating functions and exponential generating functions of these sequences. Also, we obtain some binomial identities, series and sums for them.

New Types of Distance Padovan Sequences via Decomposition Technique

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.

A STUDY ON BINOMIAL TRANSFORM OF THE GENERALIZED PADOVAN SEQUENCE

In this paper, we define the binomial transform of the generalized Padovan sequence and as special cases, the binomial transform of the Padovan, Perrin, Padovan-Perrin, modified Padovan, adjusted Padovan sequences will be introduced. We investigate their properties in details.

De Moivre-Type Identities for the Padovan Numbers

At this work, we give a method for constructing the Perrin and Padovan sequences and obtain the De Moivre-type identity for Padovan numbers. Also, we define a Padovan sequence with new initial conditions and find some identities between all of these auxiliary sequences. Furthermore, we give quadratic approximations for these sequences.

Relations on higher dimensional Padovan sequences

Many papers developed so far for Padovan sequences properties and its extensions usually follow the one-dimensional approach. The presented work introduces new relations for a higher dimensional sequence, this approach is adopted for two, three and n-dimensional Padovan Sequence. Several mathematical properties are discussed for the first time in the present work.

An extension of Lucas identity via Pascal's triangle

The Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity. An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences. This family both contains the Fibonacci and the Padovan sequences. A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.

Representation of solutions of eight systems of difference equations via generalized Padovan sequences

We indicate that the systems of difference equations $$ x_{n+1}=f^{-1}big( afleft( p_{n-1}right)+bfleft( q_{n-2}right) big) , y_{n+1}=f^{-1}big( afleft( r_{n-1}right)+bfleft( s_{n-2}right) big) , nin mathbb{N}_{0},$$ where the sequences $p_{n}$, $q_{n}$, $r_{n}$, $s_{n}$ are some of the sequences $x_{n}$ and $y_{n}$, $f : D_f longrightarrow mathbb{R}$ be a $ ``1-1 $ continuous function on its domain $D_f subseteq mathbb{R}$, initial values $x_{-j}$, $y_{-j}$, $jin{0,1,2}$ are arbitrary real numbers in $D_f$ and the parameters $a,b $ are arbitrary complex numbers, with $bneq 0$, can be solved in the closed form in terms of generalized Padovan sequences.

On eight solvable systems of difference equations in terms of generalized Padovan sequences

On eight solvable systems of difference equations in terms of generalized Padovan sequences

Determinant identities for toeplitz-hessenberg matrices with tribonacci entries

In this paper‎, ‎we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries‎. ‎These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers‎. ‎In particular‎, ‎we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants‎. ‎We then obtain‎, ‎by combinatorial arguments‎, ‎extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form $T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)}$ for $n geq r$‎, ‎with the appropriate initial conditions‎, ‎where $r geq 3$ is arbitrary‎.

On factorials in Perrin and Padovan sequences

On factorials in Perrin and Padovan sequences

On the solutions of a second-order difference equation in terms of generalized Padovan sequences

Abstract This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation x n + 1 = α x n − 1 + β γ x n x n − 1 , n ∈ N 0 , $$\begin{array}{} \displaystyle x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{array}$$ where ℕ0 = ℕ ∪ {0}, α, β, γ ∈ ℝ+, and the initial conditions x –1 and x 0 are non zero real numbers such that their solutions are associated to generalized Padovan numbers. Also, we investigate the two-dimensional case of the this equation given by x n + 1 = α x n − 1 + β γ y n x n − 1 , y n + 1 = α y n − 1 + β γ x n y n − 1 , n ∈ N 0 . $$\begin{array}{} \displaystyle x_{n+1} = \frac{\alpha x_{n-1} + \beta}{\gamma y_n x_{n-1}}, \qquad y_{n+1} = \frac{\alpha y_{n-1} +\beta}{\gamma x_n y_{n-1}} ,\qquad n\in \mathbb{N}_0. \end{array}$$

K-Step Sum and m-Step Gap Fibonacci Sequence

For two given integers k, m, we introduce the k-step sumand m-step gap Fibonacci sequence by presenting a recurrence formula that generates the nth term as the sum of k successive previous terms starting the sum at the mth previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of k, m. Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated {0,1}-matrix.

Recurrent equations of physicochemical constants for homologs substantiated with numerical sequences

The applicability of the unified first order recurrent equations A(n + 1) = aA(n) + b to the approximation of the physicochemical constants of various organic compounds (not only members of homologous series), previously found for normal boiling points, was extended to the dielectric constant (e). This tendency is equivalent to the general procedure for calculating the e of any compound from the data for preceding homologs with an accuracy comparable to the average interlaboratory reproducibility of the results of e measurements. To substantiate this universal character of recurrent relations we consider the analogy between the constants of successive homologs and the recursive numerical sequences (Fibonacci and Padovan sequences versus Lucas and Perrin sequences, respectively).

The Fibonacci-Padovan sequence and MacWilliams transform matrices

A relationship between the MacWilliams transform matrices and the classical integer Fibonacci, Lucas, and Padovan sequences is established. Namely, it is proved that the summation over some naturally chosen planes in the pyramid composed of these matrices yields a new integer sequence, which is the convolution of the Fibonacci numbers and the (alternating) Padovan numbers. In turn, this convolution is linearly represented in terms of the Lucas numbers and the Padovan numbers.

Trace maps of general Padovan lattices

The two kinds of seven-dimensional trace maps of a new class of three-component quasiperiodic lattices, which are constructed based on the general Padovan sequences S l+1 ={ S l−1 m , S l−2 n }, are derived for arbitrary integer value of m and n. It is shown that these lattices can be grouped into two distinct class. The lattices in class I correspond to n=1 and arbitrary m. They are shown to have volume-preserving second kind maps. The results are compared with those of other three-component quasiperiodic lattices.