On the Lucas triangle and its relationship with the k-Lucas numbers

Abstract

After defining the $k$-Lucas numbers of similar form to as the $k$-Fibonacci numbers are defined, a table with the polynomic expression of the first numbers of Lucas is indicated. The coefficients of these polynomials, properly placed, constitute a table that receives the name of Lucas triangle. Later, we study some properties of this triangle and the sequences obtained from this one, either are by rows, or by diagonals or antidiagonals. Finally we generate the classical Pascal trinagle from the $k$-Lucas triangle.