On the Francois numerical sequence, his board and some tillings properties Sobre a sequência numérica de François, seu Tabuleiro e a visualização de propriedades com ladrilhos

Abstract

The study of recurring numerical sequences imposes a certain restrictive and limited aspect, since, predominantly, the compendiums of History of Mathematics books emphasize only the role of the recurring numerical sequence of 2nd order, called the Fibonacci sequence. On the other hand, when we examine the modern interest in the generalization of several other cases of recurring numerical sequences, we confirm that the combinatorial approach provides the discovery of numerous mathematical properties and, especially, when we examine such properties using the notion of Chessboard. Thus, the present work addresses the case of a recurring numerical sequence recently introduced in the scientific literature, called the Francois numerical sequence. Finally, from the observation of certain cases related to the notion of hexagonal board, the work presents a formulation that, indirectly, ends up using properties related to the Lucas and Fibonacci sequences.