相关论文: Identities for Tribonacci-related sequences
We discuss an equivalence relation on the set of square binary matrices with the same number of 1's in each row and each column. Each binary matrix is represented using ordered n-tuples of natural numbers. We give a few starting values of…
In this paper, we give some properties of the Tribonacci and Tribonacci-Lucas quaternions and obtain some identities for them.
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix.…
It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary…
The classical Fibonacci sequence is known to exhibit many fascinating properties. In this paper, we explore the Fibonacci sequence and integer sequences generated by second order linear recurrence relations with positive integer…
We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and…
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 present a differential-calculus-based method which allows one to derive more identities from {\it any} given Fibonacci-Lucas identity containing a finite number of terms and having at least one free index. The method has two {\it…
We discuss an interesting sequence defined recursively; namely, sequence A105774 from the On-Line Encyclopedia of Integer Sequences, and study some of its properties. Our main tools are Fibonacci representation, finite automata, and the…
We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous…
We study the Fibonacci and Lucas numbers and demonstrate how identities can be constructed by investigating trivalent graphs and splitting fields.
We present a lovely connection between the Fibonacci numbers and the sums of inverses of $(0,1)-$ triangular matrices, namely, a number $S$ is the sum of the entries of the inverse of an $n \times n$ $(n \geq 3)$ $(0,1)-$ triangular matrix…
We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…
We present here some new identities for generalizations of Fibonacci and Lucas numbers by combinatorially interpreting these numbers in terms of numbers of certain tilings of a $1 \times m$ board. As a consequence, some new interesting…
A second order polynomial sequence is of \emph{Fibonacci-type} (\emph{Lucas-type}) if its Binet formula has a structure similar to that for Fibonacci (Lucas) numbers. Known examples of these type of sequences are: Fibonacci polynomials,…
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…
In this paper, we construct Pell matrices, analogous to Fibonacci matrices, to study algebraic properties of Pell numbers via linear algebra. This framework yields identities involving the trace, inverse, and determinant, as well as matrix…
In this paper, we define Tribonacci-Lucas polynomials and present Tribonacci-Lucas numbers and polynomials as a binomial sum. Then, we introduce incomplete Tribonacci-Lucas numbers and polynomials. In addition we derive recurrence…
We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…