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Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

Hung Viet Chu, Aney Manish Kanji, Zachary Louis Vasseur

TL;DR

Addresses the problem of characterizing integers whose Chung-Graham decomposition with even-indexed Fibonacci numbers omits a fixed term $F_{2N}$ (and its double). The authors leverage a golden-string framework and a row-based decomposition of numbers to describe the sets $A_{2k}$ and the counting structure via $N_B(n) = \lfloor (n+1)/\phi \rfloor$. They prove the main result, giving an explicit description of $B_{2N}$ as a union of arithmetic-like progressions translated by $F_{2k}$, and show how integers are organized into consecutive rows according to their largest summand. The work extends Zeckendorf-type analyses to fixed-term omissions and provides a complete fixed-term description with potential implications for combinatorial number systems and integer representations.

Abstract

As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either $0, 1$, or $2$ so that between two coefficients $2$, there must be a coefficient $0$. This paper characterizes all positive integers that do not have $F_{2k}$ ($k\ge 1$) in their decompositions. This continues the work of Kimberling, Carlitz et al., Dekking, and Griffiths, to name a few, who studied such a characterization for Zeckendorf decomposition.

Fixed-Term Decompositions Using Even-Indexed Fibonacci Numbers

TL;DR

Addresses the problem of characterizing integers whose Chung-Graham decomposition with even-indexed Fibonacci numbers omits a fixed term (and its double). The authors leverage a golden-string framework and a row-based decomposition of numbers to describe the sets and the counting structure via . They prove the main result, giving an explicit description of as a union of arithmetic-like progressions translated by , and show how integers are organized into consecutive rows according to their largest summand. The work extends Zeckendorf-type analyses to fixed-term omissions and provides a complete fixed-term description with potential implications for combinatorial number systems and integer representations.

Abstract

As a variant of Zeckendorf's theorem, Chung and Graham proved that every positive integer can be uniquely decomposed into a sum of even-indexed Fibonacci numbers, whose coefficients are either , or so that between two coefficients , there must be a coefficient . This paper characterizes all positive integers that do not have () in their decompositions. This continues the work of Kimberling, Carlitz et al., Dekking, and Griffiths, to name a few, who studied such a characterization for Zeckendorf decomposition.
Paper Structure (4 sections, 14 theorems, 67 equations)

This paper contains 4 sections, 14 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The golden string
  3. The ordered list of positive integers $n$ with $F_{2k} = \min\mathcal{CG}(n)$
  4. Proof of the main theorem

Key Result

Theorem 1.1

CG Every positive integer $n$ can be uniquely represented as a sum $n = \sum_{i\ge 1} c_i F_{2i}$, where $c_i$'s are in $\{0, 1, 2\}$ so that if $c_i = c_j = 2$ with $i < j$, then for some $k$, $i < k < j$, we have $c_k = 0$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 16 more