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The Chung-Graham Expansion

Sungkon Chang

TL;DR

This work generalizes the Chung-Graham expansion by showing that every positive integer admits a unique expansion using Fibonacci terms with equally spaced indices $F_{2+d(k-1)}$, for fixed even $d$. It defines the Chung-Graham rule with bounds $A=K_d-1$ and $B=F_{2+d}-1$, proves existence and uniqueness of expansions via a constructive induction and a lexicographic-block approach, and develops a detailed combinatorial framework of coefficient sequences and blocks. Central to the result is a maximal coefficient sequence $\beta^{(n)}$ and a block-decomposition theory that yields a well-defined, order-based construction of all feasible coefficient sequences; this leads to a bijection between coefficient sequences and positive integers through the evaluation map. The findings extend Zeckendorf-type representations to broader Fibonacci-indexed subsequences and provide a robust, constructive method for obtaining unique representations with equally-spaced Fibonacci terms, with potential applications to financial series, combinatorial encoding, and number representations.

Abstract

Chung and Graham introduced a method to uniquely represent each positive integer using even-indexed Fibonacci terms. We generalize this result to represent each positive integer using other Fibonacci terms with equally-spaced indices.

The Chung-Graham Expansion

TL;DR

This work generalizes the Chung-Graham expansion by showing that every positive integer admits a unique expansion using Fibonacci terms with equally spaced indices , for fixed even . It defines the Chung-Graham rule with bounds and , proves existence and uniqueness of expansions via a constructive induction and a lexicographic-block approach, and develops a detailed combinatorial framework of coefficient sequences and blocks. Central to the result is a maximal coefficient sequence and a block-decomposition theory that yields a well-defined, order-based construction of all feasible coefficient sequences; this leads to a bijection between coefficient sequences and positive integers through the evaluation map. The findings extend Zeckendorf-type representations to broader Fibonacci-indexed subsequences and provide a robust, constructive method for obtaining unique representations with equally-spaced Fibonacci terms, with potential applications to financial series, combinatorial encoding, and number representations.

Abstract

Chung and Graham introduced a method to uniquely represent each positive integer using even-indexed Fibonacci terms. We generalize this result to represent each positive integer using other Fibonacci terms with equally-spaced indices.

Paper Structure

This paper contains 8 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. The rule of expansion
  3. The recurrence and the maximal coefficient sequence s
  4. Proof of Theorem \ref{['thm:main']}
  5. Existence
  6. Uniqueness
  7. Construction of coefficient sequences
  8. Expansions of least upper bounds

Key Result

Theorem 2.2

Let $d$ be a positive even integer. Then, for each $n\in\mathbb{N}$, there is a unique coefficient sequence $\epsilon$ satisfying the Chung-Graham rule of expansion for even interval $d$ such that $n=\sum_{k=1}^\infty \epsilon_k F_{2+d(k-1)}$.

Theorems & Definitions (24)

  • Definition 1.2
  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 14 more