An extension of Lucas identity via Pascal's triangle

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

References

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