Related papers: A method for obtaining Fibonacci identities
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
Using a direct algebraic approach we derive convolution identities for second order sequences, hereby distinguishing between sequences obeying the same or different recurrence relations. We also state a general convolution for Horadam…
We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a…
The Lucas sequences are integers defined by a homogeneous recurrence relation. They include the well-known Fibonacci numbers, which appear abundantly in nature. The complementary Lucas numbers, defined by the same recurrence relation, are…
In this paper, we discuss about some results on average of Fibonacci and Lucas sequences that may be found in the OEIS code A111035.
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical…
In these notes we address the study of the log-concave operator acting on Lucas Sequences of first kind. We will find for which initial values a generic Lucas sequence is log-concave, and using this we show when the same sequence is…
Starting with some determinants of binomial coefficients which are related to Fibonacci and Lucas polynomials we study similar determinants for some generalizations of these polynomials and their q-analogues.
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…
The aim of this short note is to show how can be derived from the properties of fundamental interpolation polynomials some nice identities.
This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…
In this paper, we define the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials, we study recurrence relations and some properties of these polynomials
We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also…
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…
Sequences of Genocchi numbers of the first and second kind are considered. For these numbers, an approach based on their representation using sequences of polynomials is developed. Based on this approach, for these numbers some identities…
In this study, we apply the binomial transforms to Tribonacci and Tribonacci-Lucas sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we illustrate the…
We introduce a new four-parameters sequence that simultaneously generalizes some well-known integer sequences, including Fibonacci, Padovan, Jacobsthatl, Pell, and Lucas numbers. Combinatorial interpretations are discussed and many…
We derive the double recurrence $e_n = \frac{1}{2}(a_{n-1}+5b_{n-1}); f_{n} = \frac{1}{2}(a_{n-1}+b_{n-1})$ with $e_0=2;f_0=0$ for the Fibonacci numbers, leading to an extremely simple and fast implementation. Though the recurrence is…
We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant.…