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.

An extension of Lucas identity via Pascal’s triangle

Vincenzi, Giovanni
2021

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4768044
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