Jeffrey Shallit

activity

20242026

collaborators

8 papers

math.CO2026

An infinite walk in $N^{16}$ , using only unit steps, with no three collinear points

Jeffrey Shallit

Is it possible to walk to infinity, avoiding three collinear points, using as steps only the standard unit basis vectors $(0, 0, \dots, 0, 1, 0, \dots, 0)$ ? We prove this is possib…

cs.FL2026

State Complexity of Shifts of the Fibonacci Word

Delaram Moradi, Pierre Popoli, Jeffrey Shallit +1

The Fibonacci infinite word $f = (f_{i})_{i \geq 0} = 01001010 \dots$ is one of the most celebrated objects in combinatorics on words. There is a simple $5$ -state automaton tha…

cs.FL2025

The Tribonacci constant and finite automata

Jeffrey Shallit

We show that there is no automaton accepting the Tribonacci representations of $n$ and $x$ in parallel, where $ψ = 1.839 \dots$ is the Tribonacci constant, and $x= \lfloor n ψ\rfloo…

math.CO2025

On a sequence of Kimberling and its relationship to the Tribonacci word

Lubomíra Dvořáková, Edita Pelantová, Jeffrey Shallit

In 2017, Clark Kimberling defined an interesting sequence $B = 0100101100 \dots$ of $0$ 's and $1$ 's by certain inflation rules, and he made a number of conjectures about thi…

math.NT2025

Balanced Fibonacci word rectangles, and beyond

Jeffrey Shallit, Ingrid Vukusic

Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved wi…

math.NT2025

New properties of the $φ$ -representation of integers

Jeffrey Shallit, Ingrid Vukusic

We prove a few new properties of the $φ$ -representation of integers, where $φ = (1 + 5) /2$ . In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we…

math.NT2025

On the $p$ -adic valuations of values of Legendre polynomials

Max A. Alekseyev, Tewodros Amdeberhan, Jeffrey Shallit +1

We prove an explicit formula for the $p$ -adic valuation of the Legendre polynomials $P_{n} (x)$ evaluated at a prime $p$ , and generalize an old conjecture of the third author. We also…

math.CO2024

Christoffel Matrices and Sturmian Determinants

Christophe Reutenauer, Jeffrey Shallit

We discuss certain matrices associated with Christoffel words, and show that they have a group structure. We compute their determinants and show a relationship between the Zolotare…

An infinite walk in N16, using only unit steps, with no three collinear points

State Complexity of Shifts of the Fibonacci Word

The Tribonacci constant and finite automata

On a sequence of Kimberling and its relationship to the Tribonacci word

Balanced Fibonacci word rectangles, and beyond

New properties of the φ-representation of integers

On the p-adic valuations of values of Legendre polynomials

Christoffel Matrices and Sturmian Determinants

An infinite walk in $N^{16}$ , using only unit steps, with no three collinear points

New properties of the $φ$ -representation of integers

On the $p$ -adic valuations of values of Legendre polynomials