Let (T m ) m≥1 be the tribonacci sequence. We show that every integer N ≥ 1 can be written as a sum of the terms α m T m , where m runs over the set of strictly positive integers and α m (m ≥ 1) are either 1 or 0. The previous representation of N is unique if each time that we have α m = 1 then at least the two coefficients directly following α m are zero, i.e., α m+1 = α m+2 = 0.
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].
In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations: [Formula: see text] where [Formula: see text] [Formula: see text] and the initial conditions are arbitrary nonzero real numbers.
In this study, we give another formula to all the generalized tribonacci [Formula: see text] and generalized tribonacci-Lucas [Formula: see text] polynomials by using the combinatorial calculus and their sum. Then, we give the explicit formula of partial derivative of such polynomials [Formula: see text], [Formula: see text] with respect to one of those variables and describe some properties.
In this study, we denote $(t'_{n}(x))_{n\in \mathbb{N}}$ the generalized Tribonacci polynomials, which are defined by $t'_{n}(x)=x^{2}t'_{n-1}(x)+xt'_{n-2}(x)+t'_{n-3}(x), n \geqslant 4,$ with $t_{1}(x)=a, t_{2}(x)=b, t_{3}(x)=cx^{2}$ and we drive an explicit formula of $(t'_{n}(x))_{n\in \mathbb{N}}$ in terms of their coefficients $T'(n,j)$, Also, we establish some properties of $(t_{n}(x))_{n\in \mathbb{N}}$. Similarly, we study the Jacobsthal polynomials $(J_{n}(x))_{n\in \mathbb{N}}$, where $J_{n}(x)=J_{n-1}(x)+x J_{n-2}(x)+ x^{2} J_{n-3}(x), n \geqslant 4$, with $J_{1}(x)= J_{2}(x)=1, J_{3}(x)=x+1$ and describe some properties.
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