Writing about writing

• How to referee a math paper, to appear in Notices of the American Mathematical Society (4 pages).

• Writing mathematics effectively (12 pages).

Research papers

For several papers, there is a link to accompanying data or a related Mathematica package.

• Algebraic power series and their automatic complexity I: finite fields (with Manon Stipulanti and Reem Yassawi) (29 pages).   arXiv
  package: IntegerSequences | talk: slides

• Limiting density of the Fibonacci sequence modulo powers of p (with Nicholas Bragman) (20 pages).   arXiv

• A natural bijection for contiguous pattern avoidance in words (with Julia Carrigan and Isaiah Hollars), Discrete Mathematics 347 (2024) 113793 (8 pages).   arXiv | journal
  talk: slides

• The lexicographically least square-free word with a given prefix (with Siddharth Berera, Andrés Gómez-Colunga, Joey Lakerdas-Gayle, John López, Mauditra Matin, Daniel Roebuck, Noam Scully, and Juliet Whidden), The Electronic Journal of Combinatorics 30 (2023) #P3.11 (43 pages).   arXiv | journal
  talk: slides

• Magic numbers in periodic sequences (with Savinien Kreczman, Luca Prigioniero, and Manon Stipulanti), 14th International Conference on Combinatorics on Words (WORDS 2023), edited by Anna Frid and Robert Mercas, Lecture Notes in Computer Science 13899 (2023) 206–219.   arXiv | journal

• Lucas' theorem modulo p², The American Mathematical Monthly 129 (2022) 846–855.   arXiv | journal
  talk: slides

• Ultimate periodicity problem for linear numeration systems (with Émilie Charlier, Adeline Massuir, and Michel Rigo), International Journal of Algebra and Computation 32 (2022) 561–596.   arXiv | journal

• Lucas congruences for the Apéry numbers modulo p² (with Reem Yassawi and Christian Krattenthaler), Integers 21 (2021) A20 (15 pages).   arXiv | journal
  talk: slides

• Automaticity and invariant measures of linear cellular automata (with Reem Yassawi), Canadian Journal of Mathematics 72 (2020) 1691–1726.   arXiv | journal

• Avoiding 5/4-powers on the alphabet of nonnegative integers (with Manon Stipulanti), The Electronic Journal of Combinatorics 27 (2020) #P3.42 (39 pages).   arXiv | journal
  proceedings version: Avoiding 5/4-powers on the alphabet of nonnegative integers (extended abstract), 24th International Conference on Developments in Language Theory (DLT 2020), edited by Nataša Jonoska and Dmytro Savchuk, Lecture Notes in Computer Science 12086 (2020) 280–293.   journal
  computations: notebook | talk: slides

• A closed-form solution might be given by a tree: valuations of quadratic polynomials (with Leyda Almodóvar, Alyssa Byrnes, Julie Fink, Xiao Guan, Aashita Kesarwani, Gary Lavigne, Luis Medina, Victor Moll, Isabelle Nogues, Senthil Rajasekaran, and Amber Yuan), SCIENTIA, Series A: Mathematical Sciences 29 (2019) 11–28.   journal

• IntegerSequences: a package for computing with k-regular sequences, 6th International Congress on Mathematical Software (ICMS 2018), edited by James H. Davenport, Manuel Kauers, George Labahn, and Josef Urban, Lecture Notes in Computer Science 10931 (2018) 414–421.   notebook version | journal
  package: IntegerSequences | talk: slides

• Avoiding fractional powers over the natural numbers (with Lara Pudwell), The Electronic Journal of Combinatorics 25 (2018) #P2.27 (46 pages).   arXiv | journal
  data: prefixes of a/b-power-free words | package: SymbolicWords | computations: notebook | talk: slides

• A matrix generalization of a theorem of Fine, Integers 18A (2018) A18 (11 pages).   arXiv | journal
  proceedings version: Binomial coefficients, valuations, and words, 21st International Conference on Developments in Language Theory (DLT 2017), edited by Émilie Charlier, Julien Leroy, and Michel Rigo, Lecture Notes in Computer Science 10396 (2017) 68–74.   journal | MR
  talk: slides

• Periodicity in the p-adic valuation of a polynomial (with Luis Medina and Victor Moll), Journal of Number Theory 180 (2017) 139–153.   arXiv | journal | MR
  talk: slides

• p-adic asymptotic properties of constant-recursive sequences (with Reem Yassawi), Indagationes Mathematicae 28 (2017) 205–220.   arXiv | journal | MR

• Profinite automata (with Reem Yassawi), Advances in Applied Mathematics 85 (2017) 60–83.   arXiv | journal | MR

• Decision algorithms for Fibonacci-automatic words, II: related sequences and avoidability (with Chen Fei Du, Hamoon Mousavi, Luke Schaeffer, and Jeffrey Shallit), Theoretical Computer Science 657 (2017) 146–162.   journal | MR

• Infinite products arising in paperfolding (with Leyda Almodóvar, Victor Moll, Hadrian Quan, Fernando Roman, and Michole Washington), Journal of Integer Sequences 19 (2016) 16.5.1 (18 pages).   journal | MR

• What's in YOUR wallet? (with Lara Pudwell), The Mathematical Intelligencer 37 (2015) 54–60.   arXiv | journal | MR
  data: transition matrices | package: PocketChange | computations: notebook
  media coverage: Scientific American blog post by Evelyn Lamb

• p-regularity of the p-adic valuation of the Fibonacci sequence (with Luis Medina), The Fibonacci Quarterly 53 (2015) 265–271.   arXiv | journal | MR

• Automatic sets of rational numbers (with Jeffrey Shallit), International Journal of Foundations of Computer Science 26 (2015) 343–365.   arXiv | journal | MR
  proceedings version: k-automatic sets of rational numbers, 6th International Conference on Language and Automata Theory and Applications (LATA 2012), edited by Adrian-Horia Dediu and Carlos Martín-Vide, Lecture Notes in Computer Science 7183 (2012) 490–501.   journal | MR
  talk: slides

• Automatic congruences for diagonals of rational functions (with Reem Yassawi), Journal de Théorie des Nombres de Bordeaux 27 (2015) 245–288.   arXiv | journal | MR
  data: Motzkin numbers modulo p^α | package: IntegerSequences | computations: notebook | talk: slides

• Classification of automorphic conjugacy classes in the free group on two generators (with Bobbe Cooper), Algorithmic Problems of Group Theory, Their Complexity, and Applications to Cryptography, edited by Delaram Kahrobaei and Vladimir Shpilrain, Contemporary Mathematics 633 (2015) 13–40.   arXiv | journal | MR
  data: automorphic conjugacy classes | package: EquivalentWords | talk: slides

• What is … an automatic sequence?, Notices of the American Mathematical Society 62 (2015) 274–276.   journal | MR
  in the same issue: About the cover (and the cover) by Bill Casselman

• A new approach to the 2-regularity of the l-abelian complexity of 2-automatic sequences (with Aline Parreau, Michel Rigo, and Élise Vandomme), The Electronic Journal of Combinatorics 22 (2015) #P1.27 (44 pages).   arXiv | journal | MR
  talk: slides

• A characterization of p-automatic sequences as columns of linear cellular automata (with Reem Yassawi), Advances in Applied Mathematics 63 (2015) 68–89.   arXiv | journal | MR
  talk: slides

• Structure and enumeration of (3+1)-free posets (with Mathieu Guay-Paquet and Alejandro Morales), Annals of Combinatorics 18 (2014) 645–674.   arXiv | journal | MR
  proceedings version: Structure and enumeration of (3+1)-free posets (extended abstract), 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), Discrete Mathematics & Theoretical Computer Science Proceedings (2013) 253–264.   arXiv | journal

• A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences (with Doron Zeilberger), Journal of Difference Equations and Applications 20 (2014) 973–988.   arXiv | journal | MR
  packages: AutoSquared (along with sample input and output), IntegerSequences

• The rational–transcendental dichotomy of Mahler functions (with Jason Bell and Michael Coons), Journal of Integer Sequences 16 (2013) 13.2.10 (11 pages).   arXiv | journal | MR

• Growing words in the free group on two generators (with Bobbe Cooper), Illinois Journal of Mathematics 55 (2011) 417–426.   arXiv | journal | MR
  data: automorphic conjugacy classes | package: EquivalentWords | talk: slides

• Boundary growth in one-dimensional cellular automata (with Charles Brummitt), Complex Systems 21 (2012) 85–116.   arXiv | journal | MR
  packages: CellularAutomatonBoundaries, CellularAutomatonData | talk: slides

• The iterated integrals of ln(1 + xⁿ) (with Tewodros Amdeberhan, Christoph Koutschan, and Victor Moll), International Journal of Number Theory 8 (2012) 71–94.   arXiv | journal | MR

• Avoiding 3/2-powers over the natural numbers (with Jeffrey Shallit), Discrete Mathematics 312 (2012) 1282–1288.   arXiv | journal | MR

• Toward a language theoretic proof of the four color theorem (with Bobbe Cooper and Doron Zeilberger), Advances in Applied Mathematics 48 (2012) 414–431.   arXiv | journal | MR
  package: ParseWords | talk: slides

• Two binomial coefficient conjectures.   arXiv
  This is the written version of a talk I gave at Doron Zeilberger's 60th birthday conference. At some point I plan to expand it.
  talk: slides

• The number of nonzero binomial coefficients modulo p^α, Journal of Combinatorics and Number Theory 3 (2011) 15–25.   arXiv | journal | MR
  package: BinomialCoefficients | talk: slides

• Iterated primitives of logarithmic powers (with Luis Medina and Victor Moll), International Journal of Number Theory 7 (2011) 623–634.   arXiv | journal | MR

• Pattern avoidance in binary trees, Journal of Combinatorial Theory, Series A 117 (2010) 741–758.   arXiv | journal | MR
  unabridged version (with more details)
  package: TreePatterns | talk: slides

• Non-regularity of ⌊α + log_k n⌋, Integers 10 (2010) 19–23.   arXiv | journal | MR

• Counting interesting elections (with Lara Pudwell), The American Mathematical Monthly 117 (2010) 167–174.   arXiv | journal | MR

• Regularity versus complexity in the binary representation of 3ⁿ, Complex Systems 18 (2009) 367–377.   arXiv | journal | MR

• The number of walks on a regular Cayley tree (with Doron Zeilberger), The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger (2009) (4 pages).   arXiv | journal

• Bounds on the frequency of 1 in the Kolakoski word (with Elizabeth Kupin).   arXiv
  Although this paper has been cited in the literature, it is unpublished. After posting the initial version, Jean-Paul Allouche informed us of Chvátal's technical report in which he obtains better bounds using a different method.

• A natural prime-generating recurrence, Journal of Integer Sequences 11 (2008) 08.2.8 (13 pages).   arXiv | journal | MR
  talk: slides
  media coverage: Pour la Science article by Jean-Paul Delahaye   blog post by Brian Hayes   edition of The Mathematical Tourist by Ivars Peterson   blog post by Jeffrey Shallit

• Local nested structure in rule 30, Complex Systems 16 (2006) 239–258.   journal | MR
  data: right border evolution | package: BijectiveRules

Survey articles

• Known families of integer solutions of x³ + y³ + z³ = n.
  data: list of solutions

• Elliptic curves and integral solutions to A⁴ + B⁴ + C⁴ = D⁴.