Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci

TL;DR

This paper investigates generalized Zeckendorf decompositions involving Fibonacci and Tribonacci sequences, revealing recurrence relations for the number of representations of integers.

Contribution

It introduces new recurrence relations for counting representations of integers as sums or differences of Fibonacci numbers, extending Zeckendorf theory.

Findings

$B(0;k)$ follows a Tribonacci-like recurrence.

$B(n;k)$ satisfies a modified recurrence.

The work extends Zeckendorf decompositions to Tribonacci sequences.

Abstract

We study $B(n;k)$, the number of ways of writing $n$ as a sum or difference of the first $k$ Fibonacci numbers. We show that $B(0;k)$ satisfies the Tribonacci-like recurrence $B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2)$ and that $B(n;k)$ satisfies a modified version of this recurrence.