Lucas Sequences Research Articles

On $k$-generalized Lucas sequence with its triangle

In this paper, we investigate several identities of $k$-generalized Lucas numbers with $k$-generalized Fibonacci numbers. We also establish a link between generalized $s$-Lucas triangle and bi$^{s}$nomial coefficients given by the coefficients of the development of a power of $(1+x+x^{2}+\cdots+x^{s}),$ with $s \in \mathbb{N}.$

On prime factors of terms of binary recurrence sequences

We establish estimates from below for the greatest prime factor of the $n$th term of a non-degenerate binary recurrence sequence when the sequence belongs to a class of sequences which includes the Lucas sequences.

A generalization of Lucas sequence and associated identities

In this paper, we attempt to generalize Lucas sequence by generating certain number of sequences whose terms are obtained by adding the last two generated terms of the preceding sequence. Lucas sequence is obtained as a particular case of generating only one sequence. Moreover we prove some of the results which can be seen as generalized form of the results which hold for Lucas sequence. We obtain Cassini-like identity for these generalized Lucas sequences.

On the reciprocal products of generalized Fibonacci sequences

In this paper, we use the properties of error estimation and the analytic method to study the reciprocal products of the bi-periodic Fibonacci sequence, the bi-periodic Lucas sequence, and the mth-order linear recursive sequence.

On the Hurwitz-Type Zeta Function Associated to the Lucas Sequence

On the Hurwitz-Type Zeta Function Associated to the Lucas Sequence

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.

Semi-modular forms from Fibonacci–Eisenstein series

In a 2021 paper, M. Just and the second author defined a class of “semi-modular forms” on $${\mathbb {C}}\backslash {\mathbb {R}}$$ , in analogy with classical modular forms, that are “half modular” in a particular sense; and constructed families of such functions as Eisenstein-like series using symmetries related to integer partitions. Looking for further natural examples of semi-modular behavior, here we construct a family of Eisenstein-like series to produce semi-modular forms, using symmetries related to Fibonacci numbers instead of partitions. We then consider other Lucas sequences that yield semi-modular forms.

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

A q-analogue of the bi-periodic Fibonacci and Lucas sequences and Rogers–Ramanujan Identities

On the greatest common divisor ofnand thenth Fibonacci number, II

AbstractLet$\mathcal {A}$be the set of all integers of the form$\gcd (n, F_n)$, wherenis a positive integer and$F_n$denotes thenth Fibonacci number. Leonetti and Sanna proved that$\mathcal {A}$has natural density equal to zero, and asked for a more precise upper bound. We prove that$$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$for all sufficiently largex. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.

Some aspects of interchanging difference equation orders

This paper builds on Roettger and Williams’ extensions of the primordial Lucas sequence to consider some relations among difference equations of different orders. This paper utilises some of their second and third order recurrence relations to provide an excursion through basic second order sequences and related third order recurrence relations with a variety of numerical illustrations which demonstrate that mathematical notation is a tool of thought.

Generalized Pisano numbers

In this paper, we define and investigate the generalized Pisano sequences and we deal with, in detail, two special cases, namely, Pisano and Pisano–Lucas sequences. We present Binet’s formulas, generating functions and Simson’s formulas for these sequences. Moreover, we give some identities and matrices associated with these sequences. Furthermore, we show that there are close relations between Pisano and Pisano–Lucas numbers and modified Oresme, Oresme–Lucas and Oresme numbers.

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

Abstract Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.

On rank of apparition of primes in Lucas sequences

On rank of apparition of primes in Lucas sequences

On two Diophantine equations concerning Lucas sequences

On two Diophantine equations concerning Lucas sequences

On the divisibility of the rank of appearance of a Lucas sequence

Let [Formula: see text] be a Lucas sequence and, for every prime number [Formula: see text], let [Formula: see text] be the rank of appearance of [Formula: see text] in [Formula: see text], that is, the smallest positive integer [Formula: see text] such that [Formula: see text] divides [Formula: see text], whenever it exists. Furthermore, let [Formula: see text] be an odd positive integer. Under some mild hypotheses, we prove an asymptotic formula for the number of primes [Formula: see text] such that [Formula: see text] divides [Formula: see text], as [Formula: see text].

Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$

In the present paper, for a positive integer $r$, we study bi-periodic $r$-Fibonacci sequence and its family of companion sequences, bi-periodic $r$-Lucas sequence of type $s$ with $1 \leq s \leq r$, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic $r$-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.

Diophantine equations involving the bi-periodic Fibonacci and Lucas sequences

In this paper, we present new identities involving the biperiodic Fibonacci and Lucas sequences. Then we solve completely some quadratic Diophantine equations involving the bi-periodic Fibonacci and Lucas sequences.

Terms of Lucas sequences having a large smooth divisor

AbstractWe show that the $Kn$ –smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers n.

PROPERTIES OF FIBONACCI AND LUCAS SEQUENCES VIA FIBONACCI LIKE MATRICES

Some interesting identities of Fibonacci and Lucas Sequences by using Fibonacci like Matrices are discussed.

Identities relating six members of the Fibonacci family of sequences

In this paper, we prove several identities each relating a sum of products of three terms coming from different members of the Fibonacci family of sequences with a comparable sum whose terms come from three other sequences. These identities are obtained as special cases of formulas relating two linear combinations of products of three generalized Fibonacci or Lucas sequences. The latter formulas are in turn obtained from a more general generating function result for the product of three terms coming from second-order linearly recurrent sequences with arbitrary initial values. We employ algebraic arguments to establish our results, making use of the Binet-like formulas of the underlying sequences. Among the sequences for which the aforementioned identities are found include the Fibonacci, Pell, Jacobsthal and Mersenne numbers, along with their associated Lucas companion sequences.