The k-Fibonacci sequence and the Pascal 2-triangle

Abstract

The general k-Fibonacci sequence {Fk, n}n = 0∞ were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.