An extension of Lucas identity via Pascal's triangle

Year 2021, Volume: 50 Issue: 3, 647 - 658, 07.06.2021

Giuseppina Anatrıello Giovanni Vincenzi

https://doi.org/10.15672/hujms.744408

Cited By: 2

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

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