[Paper Review] The number of distinct and repeated squares and cubes in the Fibonacci sequence

Yuke Huang, Zhi‐Ying Wen|arXiv (Cornell University)|Mar 14, 2016

semigroups and automata theory9 references3 citations

TL;DR

This paper provides explicit formulas for the number of distinct and repeated squares and cubes in the Fibonacci word prefix of length $ n $, leveraging the recursive structure of return words and singular words. It derives closed-form expressions for $ A(n) $, $ B(f_m) $, $ C(n) $, and $ D(f_m) $, and presents efficient algorithms to compute $ B(n) $ and $ D(n) $ for any $ n $, extending prior results on Fibonacci word combinatorics.

ABSTRACT

The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $σ(a,b)=(ab,a)$. In this paper, we get the explicit expressions of all squares and cubes, then we determine the number of distinct squares and cubes in $\mathbb{F}[1,n]$ for all $n$, where $\mathbb{F}[1,n]$ is the prefix of $\mathbb{F}$ of length $n$. By establishing and discussing the recursive structure of squares and cubes, we give algorithms for counting the number of repeated squares and cubes in $\mathbb{F}[1,n]$ for all $n$, and get explicit expressions for some special $n$ such as $n=f_m$ (the Fibonacci number) etc., which including some known results such as in A.S.Fraenkel and J.Simpson, J.Shallit et al.

Motivation & Objective

• To determine the number of distinct and repeated squares and cubes in the Fibonacci word prefix $ \mathbb{F}[1,n] $ for all $ n $.
• To extend known results on square and cube occurrences in the Fibonacci sequence, particularly those by Fraenkel, Simpson, and Shallit et al.
• To develop efficient algorithms for computing the number of repeated squares $ B(n) $ and repeated cubes $ D(n) $, even when no closed-form exists.
• To establish a recursive framework using return words and singular words to analyze the distribution of squares and cubes in the Fibonacci sequence.

Proposed method

• Utilizes the structure of return words in the Fibonacci sequence, where the sequence of return words for any factor forms a Fibonacci word over two return symbols.
• Applies Property 2.2, which states that the return word sequence for any factor is itself a Fibonacci word, enabling recursive analysis of square and cube occurrences.
• Employs singular words $ K_m $, which are palindromic and maximal in their occurrence, to characterize the kernel of any factor and its return structure.
• Derives recursive relations for square and cube positions using the return word structure, allowing efficient computation of counts via dynamic decomposition.
• Uses the identity $ f_m = |F_m| $, the Fibonacci number, to define key intervals for recursive reduction of $ D(n) $ and $ B(n) $.
• Applies Corollary 10.6 to compute partial sums $ \sum_{i=f_m-1}^n d(i) $ efficiently by reducing the problem to smaller intervals based on Fibonacci number identities.

Experimental results

Research questions

• RQ1 What is the exact number of distinct squares $ A(n) $ in the Fibonacci word prefix of length $ n $, and can it be expressed in closed form?
• RQ2 How can the number of repeated squares $ B(n) $ be computed efficiently for any $ n $, even without a closed-form expression?
• RQ3 What is the explicit formula for the number of distinct cubes $ C(n) $ in $ \mathbb{F}[1,n] $, and how does it relate to the Fibonacci structure?
• RQ4 What is the closed-form expression for the number of repeated cubes $ D(f_m) $ at Fibonacci-length prefixes, and how does it compare to prior results?
• RQ5 Can a fast algorithm be constructed to compute $ D(n) $ for arbitrary $ n $, using recursive reduction based on Fibonacci number decomposition?

Key findings

• The number of distinct squares $ A(n) $ in $ \mathbb{F}[1,n] $ is explicitly determined for all $ n $, providing a complete characterization of square occurrences.
• The number of repeated squares $ B(f_m) $ is given by $ \frac{m-11}{5}f_{m-1} + \frac{m+1}{5}f_{m-3} + m + 1 $ for $ m \geq 6 $, extending prior work by Fraenkel and Simpson.
• The number of distinct cubes $ C(n) $ is determined via recursive structure, with $ C(n) $ expressible in terms of Fibonacci numbers and return word properties.
• The number of repeated cubes $ D(f_m) $ is explicitly given by $ \frac{m-11}{5}f_{m-1} + \frac{m+1}{5}f_{m-3} + m + 1 $ for $ m \geq 6 $, matching and re-expressing the result from Shallit et al. using algebraic identities.
• A fast algorithm is constructed for computing $ D(n) $ for any $ n $, using recursive reduction via Corollary 10.6 and partial sum computation over intervals defined by Fibonacci numbers.
• The paper confirms the equivalence of its expression for $ D(f_m) $ with the exponential form from Shallit et al., validating the new algebraic derivation through exact coefficient comparison.