- Hacettepe Journal of Mathematics and Statistics
- Vol: 50 Issue: 3
- An extension of Lucas identity via Pascal's triangle
An extension of Lucas identity via Pascal's triangle
Authors : Giuseppina Anatriello, Giovanni Vincenzi
Pages : 647-658
Doi:10.15672/hujms.744408
View : 10 | Download : 4
Publication Date : 2021-06-07
Article Type : Research
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