Zeckendorf's Theorem Using Indices in an Arithmetic Progression

TL;DR

This paper generalizes Zeckendorf's Theorem by proving that integers can be uniquely decomposed into Fibonacci numbers with indices in any arithmetic progression, introducing new Fibonacci recurrences.

Contribution

It extends Zeckendorf's Theorem to indices in arbitrary arithmetic progressions and derives new Fibonacci recurrences.

Findings

Unique decompositions using Fibonacci indices in arithmetic progressions

New Fibonacci recurrences of independent interest

Generalization of existing Fibonacci decomposition results

Abstract

Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their own right.