An extension of Lucas identity via Pascal`s triangle

Authors : Giuseppina ANATRIELLO, Giovanni VİNCENZİ

Pages : 647-658

Doi:10.15672/hujms.744408

View : 13 | Download : 5

Publication Date : 2021-06-07

Article Type : Research Paper

Abstract :The Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity. An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences. This family both contains the Fibonacci and the Padovan sequences. A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.
Keywords : combinatorial identities, Pascal, k Fibonacci diagonals, Fibonacci sequence, Padovan sequence