A Novel Proof for Kimberling's Conjecture on Doubly Fractal Sequences

TL;DR

This paper introduces a new method for constructing doubly fractal sequences and proves Kimberling's conjecture that signature sequences are exactly the doubly fractal sequences.

Contribution

It provides a constructive procedure for doubly fractal sequences and confirms Kimberling's conjecture, advancing understanding of fractal sequences in mathematics.

Findings

Proved Kimberling's conjecture on doubly fractal sequences

Developed a new construction procedure for these sequences

Established the equivalence between signature and doubly fractal sequences

Abstract

A sequence is a fractal sequence if it contains itself as a proper subsequence. (The self-containment property resembles that of visual fractals) A doubly fractal sequence of integers is defined by operations called upper trimming and lower trimming. C. Kimberling proved that signature sequences are doubly fractal and conjectured the converse. This article gives a procedure for constructing doubly fractal sequences and proves Kimberling's conjecture.