Pell Lucas Sequences Research Articles

GENERALIZATION OF PELL SEQUENCE AND PELL-LUCAS SEQUENCE, NEW RESULTS AND CIRCULANT MATRICES ASPECTS

Some generalizations of Pell sequence and Pell-Lucas sequence, namely, (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence are considered in this paper. We obtain generating functions, some identities and formulas for the sums of a finite number of terms and consecutive terms, sums of squares of consecutive terms and alternating sums of consecutive terms of these sequences. Then, we obtain the eigenvalues and determinants of particular circulant matrices involving (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence. Finally, we obtain some upper and lower bounds for the spectral norms of these circulant matrices.

On Mixed 𝐵-Concatenations of Pell and Pell–Lucas Numbers which are Pell Numbers

Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.

On Perfect Powers in $$k$$-Generalized Pell–Lucas Sequence

On Perfect Powers in $$k$$-Generalized Pell–Lucas Sequence

Identities of Generalized Pell and Pell-Lucas Sequences

In this paper, we present sums of generalized Pell and Pell-Lucas sequences. These sequences were introduced by Tulay Yagmur in 2019. We establish some connection formulae of involving them. Also, we present its two cross two matrix representation. We have used their Generating function, Binet’s formula and Induction method to derive the identities.

On generalized (k, r)-Pell and (k, r)-Pell–Lucas numbers

We introduce new kinds of k-Pell and k-Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k,r)-Pell and (k,r)-Pell–Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k,r)-Pell and (k,r)-Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k,r)-Pell and (k,r)-Pell–Lucas sequences from the generating function, in connection to graphs.

An exponential Diophantine equation involving the sum or difference of powers of two Pell numbers

Abstract Let P k and Q k be the k-th terms of the Pell and Pell-Lucas sequences respectively. In this paper, we find all the solutions of the Diophantine equations P n + 1 x ± P n − 1 x = Q m $$\begin{array}\displaystyleP_{n+1}^x\pm P_{n-1}^x=Q_m\end{array}$$ in nonnegative integer variables (m, n, x).

Solutions of equations [formula omitted

In the present paper, we have solved the equation x2−(p2q2±3p)y2=kt,x2−(p2q2±5p)y2=kt and expressed its positive integer solutions in terms of generalized Fibonacci, generalized Lucas and generalized Pell, generalized Pell–Lucas sequences. With the help of this equation, we have found units of Z[885] and Z[915] in terms of generalized Fibonacci, generalized Lucas, generalized Pell and generalized Pell–Lucas numbers.

A Note on Two Fundamental Recursive Sequences

Abstract In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first n terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number e(≈ 2:71828:::).

On Generalized Lucas Pseudoprimality of Level k

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.

On some new results for the generalised Lucas sequences

Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

On Some New Arithmetic Properties of the Generalized Lucas Sequences

Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.

On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences

On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences

Pell and Pell–Lucas numbers of the form $$x^a\pm x^b+1$$

Let $$P_k$$ and $$Q_k$$ be the $$k^{\text{ th }}$$ Pell and Pell–Lucas terms of the Pell sequence $$\{P_n\}_{n\ge 0}$$ and the Pell–Lucas sequence $$\{Q_n\}_{n\ge 0}$$ , respectively. In this paper, we study the Diophantine equations $$P_n = x^a\pm x^b+1$$ and $$Q_n = x^a\pm x^b+1$$ , in positive integers (n, x, a, b) and determine the explicit upper bounds for n. We also completely solve these equations in positive integers (n, x, a, b) with $$0\le b < a$$ and $$2\le x \le 20$$ .

Perfect Pell and Pell–Lucas numbers

Abstract The Pell sequence is given by the recurrence Pn = 2Pn−1 + Pn−2 with initial condition P0 = 0, P1 = 1 and its associated Pell-Lucas sequence is given by the same recurrence relation but with initial condition Q0 = 2, Q1 = 2. Here we show that 6 is the only perfect number appearing in these sequences. This paper continues a previous work that searched for perfect numbers in the Fibonacci and Lucas sequences.

A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence)

In this study, we bring into light, a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as \begin{align*} Q_{n}= \begin{cases} 2bQ_{n-1}+Q_{n-2},&\text{if} n \text{is even} 2aQ_{n-1}+Q_{n-2},&\text{if} n \text{is odd}% \end{cases} \text{ }n\geq 2, \end{align*} with initial conditions \(Q_{0}=2\), \(Q_{1}=a\). The Binet formula as well as the generating function for this sequence are given. The convergence properties of the consecutive terms of this sequence are also examined after which the well known Cassini, Catalans and the D'Ocagne's identities as well as some related summation formulas are also given.

Gaussian (𝒔, 𝒕)-Pell and Pell-Lucas Sequences and Their Matrix Representations

In this study, we define the Gaussian (s,t)-Pell and Gaussian (s,t)-Pell-Lucas sequences. Then, by using these sequences we define Gaussian (s,t)-Pell and Gaussian (s,t)-Pell-Lucas matrix sequences. Thereafter, we give generating functions, Binet’s formulas and some summation formulas of these sequences. Finally, we obtain some relationships between Gaussian (s,t)-Pell and Gaussian (s,t)-Pell-Lucas matrix sequences.

(s, t)-PELL VE (s, t)-PELL-LUCAS MATRİS DİZİLERİ ÜZERİNE

Number sequence matrices is a widely studied subject in matrix analysis. Especially number sequence matrices whose entries are well-known number sequences have become a very interesting research subject in recent years. We have seen many studies on the different number sequences in the last years. Fibonacci and Lucas number sequences are the best of these number sequences. In this sequences each term is the sum of two previous terms, with initial values F00, F1 and L2, L respectively. In Pell and Pell-Lucas number 0 1F 1L sequences, nth term of the sequence is equal to the sum of n-2 th term and two times n-1 th term. In literature, many proporties belong to number and matrix sequences constructed by recursion relations like these sequences. In this study, we present some important relationships between s, t -Pell and s, t -Pell-Lucas matrix sequences. Some identities for s, t -Pell and s, t -Pell-Lucas sequences are obtained by using these matrix sequences. Furthermore, we give the Binet Formulas for nth s, t -Pell and s, t -Pell-Lucas sequences. And in this formulas we will determine some relations between s, t -Pell and s, t -Pell-Lucas sequences

On the determination of solutions of simultaneous Pell equations $$x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1$$ x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

In this paper, we consider the simultaneous Pell equations 0.1 $$\begin{aligned} x^{2}-(a^{2}-1)y^{2}= & {} 1, \nonumber \\ y^{2}-pz^{2}= & {} 1, \end{aligned}$$ where p is prime and $$a>1.$$ Assuming the solutions of the Pell equation $$ x^{2}-(a^{2}-1)y^{2}=1$$ are $$x=x_{m}$$ and $$y=y_{m}$$ with $$m\ge 2,$$ we prove that the system (0.1) has solutions only when $$m=2$$ or $$m=3.$$ In the case of $$m=3,$$ we show that $$p=2$$ and give the solutions of (0.1) in terms of Pell and Pell–Lucas sequences. When $$m=2$$ and $$p\equiv 3(\hbox {mod}\, 4),$$ we determine the values of a, x, y, and z. Lastly, we show that (0.1) has no solutions when $$p\equiv 1(\hbox {mod}\,4)$$ .

On polynomial identities for recursive sequences

In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of obtained expressions to the other recursive sequences.

Generalizations of Modified Pell and Pell-Lucas Sequences and Their Generating Matrices and Some Sums

This paper deals with the generalizations of the usual Modified Pell numbers and Pell-Lucas and an investigation of their certain identities. For this purpose, the generalized Modified Pell numbers and Pell-Lucas numbers are created, and appropriate initial conditions for them are presented. Some sum formulae consisting of the generalized Modified Pell numbers and Pell-Lucas numbers are investigated. A very fundamental features corresponding to these new generalizations are obtained by using matrix representations. Especially, an important relationship between the generalized Modified Pell and Pell-Lucas numbers is discovered.