Graphical analysis of the movements of a line-following robot

The difficulties in the interpretation of graphs in kinematics have been studied since the 1980s. To try to reduce students’ learning problems, the researchers point to the need for different didactic strategies, some with the use of new technologies. In this context, we present an experimental proposal that uses a line-following robot based on Arduino to teach graph interpretation in movements with constant velocity.

Distance Fibonacci Polynomials by Graph Methods

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

On $$F_3(k,n)$$-numbers of the Fibonacci type

In this paper, we study a generalization of Narayana’s numbers and Padovan’s numbers. This generalization also includes a sequence whose elements are Fibonacci numbers repeated three times. We give combinatorial interpretations and a graph interpretation of these numbers. In addition, we examine matrix generators and determine connections with Pascal’s triangle.

Distance Fibonacci Polynomials—Part II

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas polynomials.

(2, k)-Distance Fibonacci Polynomials

In this paper we introduce and study (2,k)-distance Fibonacci polynomials which are natural extensions of (2,k)-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of (2,k)-distance Fibonacci polynomials with Pascal’s triangle.

On Some Multinomial Sums Related to the Fibonacci Type Numbers

In this paper we investigate Fibonacci type sequences defined by kth order linear recurrence. Based on their companion matrix and its graph interpretation we determine multinomial and binomial formulas for these sequences. Moreover we present a graphical rule for calculating the words of these sequences from the Pascal’s triangle.

Distance Fibonacci Polynomials

In this paper, we introduce a new kind of generalized Fibonacci polynomials in the distance sense. We give a direct formula, a generating function and matrix generators for these polynomials. Moreover, we present a graph interpretation of these polynomials, their connections with Pascal’s triangle and we prove some identities for them.