On a family of recurrences that includes the Fibonacci and the Narayana recurrences

TL;DR

This paper explores a family of recurrence relations, including Fibonacci and Narayana sequences, analyzing their properties and connections to various mathematical concepts like integer representations, Pascal's triangle, and Nim games.

Contribution

It provides a comprehensive survey and proofs of properties for this family of recurrences, linking them to multiple areas in combinatorics and number theory.

Findings

Identifies key properties of the recurrence family

Establishes connections to Pascal's triangle and Nim games

Provides proofs for various recurrence-related properties

Abstract

We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.