Alternate Lucas Cubes

We introduce alternate Lucas cubes, a new family of graphs designed as an alternative for the well known Lucas cubes. These interconnection networks are subgraphs of Fibonacci cubes and have a useful fundamental decomposition similar to the one for Fibonacci cubes. The vertices of alternate Lucas cubes are constructed from binary strings that are encodings of Lucas representation of integers. As well as ordinary hypercubes, Fibonacci cubes and Lucas cubes, alternate Lucas cubes have several interesting structural and enumerative properties. In this paper we study some of these properties. Specifically, we give the fundamental decomposition giving the recursive structure, determine the number of edges, number of vertices by weight, the distribution of the degrees; as well as the properties of induced hypercubes, [Formula: see text]-cube polynomials and maximal hypercube polynomials. We also obtain the irregularity polynomials of this family of graphs, determine the conditions for Hamiltonicity, and calculate metric properties such as the radius, diameter, and the center.

Representation of integers by k-generalized Fibonacci sequences and applications in cryptography

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

Representation of integers: a nonclassical point of view

In \cite{A.Boudaoud1}, the author asked the following question: Which $n\in \mathbb{N}$ unlimited can be represented as a sum $% n=s+w_{1}w_{2}$, where $s\in \mathbb{Z}$ is a limited integer and $\omega _{1}$, $\omega _{2}$ are two unlimited positive integers? In this survey article we partially answer this question, i.e., we present some families of unlimited positive integers which can be written as the sum of a limited integer and the product of at least two unlimited positive integers.

Representation of Integers as Sums of Fibonacci and Lucas Numbers

Motivated by the Elementary Problem B-416 in the Fibonacci Quarterly, we show that, given any integers n and r with n≥2, every positive integer can be expressed as a sum of Fibonacci numbers whose indices are distinct integers not congruent to r modulo n. Similar expressions are also dealt with for the case of Lucas numbers. Symmetric and anti-symmetric properties of Fibonacci and Lucas numbers are used in the proofs.