Sergio Falcon

Mesh quality improvement and other properties in the four-triangles longest-edge partition

The four-triangles longest-edge (4T-LE) partition of a triangle t is obtained by joining the midpoint of the longest edge of t to the opposite vertex and to the midpoints of the two remaining edges. The so-called self-improvement property of the refinement algorithm based on the 4-triangles longest-edge partition is discussed and delimited by studying the number of dissimilar triangles arising from the 4T-LE partition of an initial triangle and its successors. In addition, some geometrical properties such as the number of triangles in each similarity class per mesh level and new bounds on the maximum of the smallest angles and on the second largest angles are deduced.

On k-Fibonacci numbers of arithmetic indexes

In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say an þ r for fixed integers a and r. This enables us to give in a straightforward way several formulas for the sums of such numbers.

Binomial Transforms of the k-Fibonacci Sequence

In this paper, we apply the binomial, k-binomial, rising, and falling transforms to the k-Fibonacci sequence. Many formulas relating the so obtained new sequences are presented and proved. Finally, we define and find the inverse transforms of the sequences previously obtained.

On the generating matrices of the Ê-Fibonacci numbers

In this paper we define some tridiagonal matrices depending of a parameter from which we will find the k-Fibonacci numbers. And from the cofactor matrix of one of these matrices we will prove some formulas for the k-Fibonacci numbers differently to the traditional form. Finally, we will study the eigenvalues of these tridiagonal matrices. Keyword : k-Fibonacci numbers, Cofactor matrix, Eigenvalues.

The k-Fibonacci matrix and the Pascal matrix

We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.

Original article: A local refinement algorithm for the longest-edge trisection of triangle meshes

Abstract: In this paper we present a local refinement algorithm based on the longest-edge trisection of triangles. Local trisection patterns are used to generate a conforming triangulation, depending on the number of non-conforming nodes per edge presented. We describe the algorithm and provide a study of the efficiency (cost analysis) of the triangulation refinement problem. The algorithm presented, and its associated triangle partition, afford a valid strategy to refine triangular meshes. Some numerical studies are analysed together with examples of applications in the field of mesh refinement.

On the complex k-Fibonacci numbers

We first study the relationship between the k-Fibonacci numbers and the elements of a subset of . Later, and since generally studies that are made on the Fibonacci sequences consider that these numbers are integers, in this article, we study the possibility that the index of the k-Fibonacci number is fractional; concretely, . In this way, the k-Fibonacci numbers that we obtain are complex. And in our desire to find integer sequences, we consider the sequences obtained from the moduli of these numbers. In this process, we obtain several integer sequences, some of which are indexed in The Online Enciplopedy of Integer Sequences (OEIS).

Combinatorial proofs of Honsberger-type identities

In this article, we consider some generalizations of Fibonacci numbers. We consider k-Fibonacci numbers (that follow the recurrence rule F k,n + 2 = kF k,n + 1 + F k,n ), the (k,ℓ)-Fibonacci numbers (that follow the recurrence rule F k,n + 2 = kF k,n + 1 + ℓF k,n ), and the Fibonacci p-step numbers ( , with , and p > 2). Then we provide combinatorial interpretations of these numbers as square and domino tilings of n-boards, and by easy combinatorial arguments Honsberger identities for these Fibonacci-like numbers are given. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem.

Classroom note: A simple proof of an interesting Fibonacci generalization

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of ‘precious metal ratios’.

Fibonacci's multiplicative sequence

The aim of this work is to consider a sequence in which each term is obtained by multiplying both previous terms. This sequence is similar to Fibonaccis sequence but with some particularities that will be proved and verified.