Walnut Papers and Books

1. Jean-Michel Couvreur, Martin Delacourt, Nicolas Ollinger, Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti, Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences, ArXiv preprint arXiv:2504.13584 [cs.FL].
2. Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arxiv preprint arXiv:2503.19696 [math.CO], March 25 2025.
3. Henk Bruin and Robbert Fokkink, On the records and zeros of a deterministic random walk, arXiv:2503.11734 [math.DS], March 14 2025.
4. Rob Burns, Chung-Graham and Zeckendorf representations, arXiv:2502.18870 [math.NT], February 26 2025.
5. Wieb Bosma, René Bruin, Robbert Fokkink, Jonathan Grube, Anniek Reuijl, Thian Tromp, Using Walnut to solve problems from the OEIS, Arxiv preprint arXiv:2503.04122 [math.NT], March 6 2025.
6. Jeffrey Shallit, The Narayana morphism and related words, ArXiv preprint arXiv:2503.01026 [math.CO], March 2 2025. Files for the paper:
   • naranum.tar, uncompress and put in the "Custom Bases" directory of Walnut
   • naracode.tar, uncompress and put in the "Automata Library" directory of Walnut
7. Jeffrey Shallit, An 'Experimental Mathematics' Approach to Stolarsky Interspersions via Automata Theory, ArXiv preprint arXiv:2502.03312 [cs.FL], February 5 2025. Available at https://arxiv.org/abs/2502.03312.
   • automata for the paper
8. Narad Rampersad and Jeffrey Shallit, Rudin-Shapiro sums via automata theory and logic, Theor. Comput. Sys. 69 (2025), Article 9.
   • Walnut code for the paper
   • RS4.txt; put this in the Word Automata Library of Walnut
   • rss.txt; put this in the Automata Library of Walnut
   • rst.txt; put this in the Automata Library of Walnut
9. G. Joshi and D. Rust, Monochromatic arithmetic progressions in the Fibonacci word, Arxiv preprint arXiv:2501.05830 [math.DS], January 10 2025.
10. Jeffrey Shallit, The Hurt-Sada array and Zeckendorf representations, arxiv preprint arXiv:2501.08823 [math.NT], January 15 2025. To verify the results in the paper, uncompress the following file and put the four automata in the Automata Library directory of Walnut:
    • wal.tar
    You can also cut-and-paste from the following file of all the Walnut commands in the paper.
    • sada-walnut2.txt
11. Jeffrey Shallit, Cloitre's self-generating sequence, arXiv preprint arXiv:2501.00784 [math.CO], January 1 2025. To verify the results in the paper, put the following automata in the Automata Library directory of Walnut:
    • e.txt
    • ep.txt
    • g.txt
    and this one in the Word Automata Library:
    • Q.txt
12. Lucas Mol, Narad Rampersad, Jeffrey Shallit, Dyck words, pattern avoidance, and automatic sequences, Communications in Mathematics 33 (2025) (2), Paper no. 5.
13. J. Shallit, Runs in paperfolding sequences, ArXiv preprint arXiv:2412.17930 [math.CO], December 23 2024. To verify the results with Walnut, put the following automata in the Automata Library of Walnut.
    • sp.txt
    • assoc.txt
14. Bastien Mignoty, Antoine Renard, Michel Rigo, and Markus A. Whiteland, Automatic proofs in combinatorial game theory, preprint, 2024.
15. Jonathan Andrade and Lucas Mol, Avoiding abelian and additive powers in rich words, Arxiv preprint arXiv:2408.15390 [math.CO], August 27 2024.
16. Gabriele Fici, Jeffrey Shallit, Jamie Simpson, Some remarks on palindromic periodicities, ArXiv preprint arXiv:2407.10564 [math.CO], July 15 204. Available at https://arxiv.org/abs/2407.10564.
17. Michel Rigo, Manon Stipulanti, and Markus A. Whiteland, Characterizations of families of morphisms and words via binomial complexities, Europ. J. Combinatorics 118 (2024), 103932.
18. Jonathan Andrade, Avoiding additive powers in words, B. Sci. thesis, Department of Mathematics and Statistics, Thompson Rivers University, 2024. Available at https://arcabc.ca/islandora/object/tru%3A6415/datastream/PDF/view.
19. Nicolas Ollinger, Jeffrey Shallit, The repetition threshold for Rote sequences, Arxiv preprint arXiv:2406.17867 [math.CO], June 25 2024. Available at https://arxiv.org/abs/2406.17867. There is some supplementary material, including
    • An html file with more information about the computations
    • A Jupyter notebook explaining what we did
20. Olivier Carton, Jean-Michel Couvreur, Martin Delacourt, Nicolas Ollinger, Addition in Dumont-Thomas Numeration Systems in Theory and Practice, Arxiv preprint arXiv:2406.09868 [cs.FL], June 14 2024. Available at https://arxiv.org/abs/2406.09868.
21. Jean-Paul Allouche, John M. Campbell, Shuo Li, Jeffrey Shallit, Manon Stipulanti, The reflection complexity of sequences over finite alphabets, Arxiv preprint arXiv:2406:09302 [math.CO], June 13 2024. Available at https://arxiv.org/abs/2406.09302. Current version is dated May 6 2025.
22. J.-P. Allouche, N. Rampersad, and J. Shallit, Repetition threshold for binary automatic sequences, Arxiv preprint arXiv:2406.06513 [math.CO], June 10 2024. Available at https://arxiv.org/abs/2406.06513. To check the claims of the paper, you can use the following Walnut code.
    • Code for checking that the morphisms in Table 1 generate squarefree words by iteration: ARS-table1.txt
    • Code for checking cases k = 3,6 in Table 2: ARS-table2.txt
    • Fibonacci DFAO in Figure 1 to compute the sequence y, put this in the Word Automata directory of Walnut: Y2.txt
    • Code for checking squarefreeness of y and computing h(y) in Remark 12: ARS-remark12.txt
    • Fibonacci DFAO for the word w in Remark 12, put this in the Word Automata directory of Walnut: W.txt
    • Walnut code checking (7/3)-freeness of w: ARS-remark12b.txt
    • Tribonacci DFAO for the word z in Theorem 13, put this in the Word Automata directory of Walnut: Z.txt
    • Walnut code for the DFAO for h(z): ARS-thm13.txt
23. Rob Burns, Synchronisation of running sums of automatic sequences, Arxiv preprint arXiv:2405.17536 [math.NT], May 27 2024. Available at https://arxiv.org/abs/2405.17536.
24. Aaron Barnoff, Curtis Bright, and Jeffrey Shallit, Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals, Arxiv preprint arXiv:2405.02727 [cs.FL], May 4 2024. Available at https://arxiv.org/abs/2405.02727.
25. Marieh Jahannia and Manon Stipulanti, Exploring the Crochemore and Ziv-Lempel factorizations of some automatic sequences with the software Walnut, Arxiv preprint arXiv:2403.15215 [cs.DM], March 22 2024. Available at https://arxiv.org/abs/2403.15215.
26. Jason Bell, Chris Schulz, and Jeffrey Shallit, Consecutive Power Occurrences in Sturmian Words, Arxiv preprint arXiv:2402.09597 [math.CO], February 14 2024. Available at https://arxiv.org/abs/2402.09597.
27. Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, Arxiv preprint arXiv:2402.08331 [math.NT], February 13 2024. Available at https://arxiv.org/abs/2402.08331.
28. Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv preprint arXiv:2312.11706 [math.CO], December 18 2023. Available at https://arxiv.org/abs/2312.11706.
29. J. Shallit and X. Xu, Repetition factorization of automatic sequences, Arxiv preprint arXiv:2311.14961 [cs.FL], November 25 2023. Available at https://arxiv.org/abs/2311.14961.
30. J.-P. Allouche and J. Shallit, Additive properties of the evil and odious numbers and similar sequences, Funct. Approx. Comment. Math. (2023), 1-15.
31. J. Shallit, A. M. Shur, and S. Zorcic, New constructions for 3-free and 3⁺-free binary morphisms. Arxiv preprint arXiv:2310.15064 [math.CO], October 23 2023.
    Walnut files for the paper:
    • Walnut commands for section 3 of the paper
    • Walnut commands for section 4 of the paper
    • Walnut commands for section 5 of the paper
    • X.txt, put in the Word Automata directory of Walnut
    • Y.txt, put in the Word Automata directory of Walnut
    • tar file of automata produced, unpack and put in the Automata Library directory of Walnut. These will be produced by the commands above if you have enough RAM on your machine and time to devote (about two weeks of computation time on a machine with 400G of RAM). But if you do not have this time you can just work with the automata produced.
32. J. Shallit, Proof of Irvine's conjecture via mechanized guessing, Arxiv preprint arXiv:2310.14252 [math.CO], October 22 2023.
    Files for the paper:
    • K.txt, put this in the "Word Automata Library" directory of Walnut.
    • files for numeration system, unpack and put both files in the "Custom Bases" directory of Walnut.
    • files for the sequences, unpack and put all these files in the "Automata Library" of Walnut.
33. J. Shallit, Proving Results About OEIS Sequences with Walnut, in C. Dubois and M. Kerber, eds., CICM 2023, LNAI Vol. 14101, Springer, 2023, pp. 270-282.
34. Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, Arxiv preprint arXiv:2309.04012 [math.NT], September 7 2023. Appeared in INTEGERS 24 (2024), Paper #A73; https://math.colgate.edu/~integers/y73/y73.pdf.
35. L. Schaeffer and J. Shallit, The first-order theory of binary overlap-free words is decidable, Canad. J. Math. (2023).
    • There is a small error in the statement of Theorem 5.5. It refers to all overlap-free words, but the theorem is actually about all overlap-free Restivo words.
    • The definition for `normalize' was omitted. You can download it here: normalize.txt.
    • The definition for `CODE' was omitted. You can download it here: CODE.txt.
36. J. Shallit, Proving properties of some greedily-defined integer recurrences via automata theory. Arxiv preprint arXiv:2308.06544 [cs.DM], August 12 2023, available at https://arxiv.org/abs/2308.06544.
37. M. Rigo, M. Stipulanti, and M. A. Whiteland, Automaticity and Parikh-collinear morphisms, in A. Frid and R. Mercas, eds., WORDS 2023, LNCS 13899, Springer, 2023, pp. 247-260.
38. J. Shallit, Proving Properties of ϕ-Representations with the Walnut Theorem-Prover, arxiv preprint arXiv:2305.02672 [math.NT], May 4 2023. Available at https://arxiv.org/abs/2305.02672.
    • All of the automata needed
    • Maple files for matrices
39. J. Shallit and A. Zavyalov, Transduction of automatic sequences and applications, arxiv preprint arXiv:2303.15203 [cs.FL], March 27 2023. Available at https://arxiv.org/abs/2303.15203.
40. Jeffrey Shallit, Rarefied Thue-Morse Sums Via Automata Theory and Logic, Arxiv preprint ArXiv:2302.09436 [math.NT], February 18 2023. Final version in J. Number Theory 257 (2024) 98-111. Available at https://doi.org/10.1016/j.jnt.2023.10.015.
    • f30.txt -- f30.txt, put this in the "Automata Library" directory of Walnut
    • f31.txt -- f31.txt, put this in the "Automata Library" directory of Walnut
    • f32.txt -- f32.txt, put this in the "Automata Library" directory of Walnut
    • f50.txt -- f50.txt, put this in the "Automata Library" directory of Walnut
    • f51.txt -- f51.txt, put this in the "Automata Library" directory of Walnut
    • conv55.txt -- conv55.txt, put this in the "Automata Library" directory of Walnut
    • moser-walnut-cmds.txt, all the Walnut commands in the paper
41. Jeffrey Shallit, Prefixes of the Fibonacci word, Arxiv preprint arXiv:2302.04640 [cs.FL], February 9 2023.
42. Jeffrey Shallit, A Dombi counterexample with positive lower density, Arxiv preprint arXiv:2302.02138 [math.NT], February 4 2023. Final revised version appeared in INTEGERS 23 (2023), #A74, and available here.
43. Narad Rampersad and Jeffrey Shallit, Rudin-Shapiro sums via automata theory and logic, Arxiv preprint arXiv:2302.00405 [math.NT], February 1 2023.
    • Walnut code for the paper
    • RS4.txt; put this in the Word Automata Library of Walnut
    • rss.txt; put this in the Automata Library of Walnut
    • rst.txt; put this in the Automata Library of Walnut
    An abbreviated version of this paper appeared in A. Frid and R. Mercas, eds., WORDS 2023, LNCS 13899, Springer, 2023, pp. 233-246.
44. Jeffrey Shallit, Proof of a conjecture of Krawchuk and Rampersad, arxiv preprint arXiv:2301.11473 [math.CO], January 27 2023. Available at https://arxiv.org/abs/2301.11473.
45. Jeffrey Shallit, Counterexample to a Conjecture of Dombi in Additive Number Theory, Arxiv preprint arXiv:2212.12473 [math.NT], December 23 2022.
46. R. Fokkink, G. F. Ortega, and D. Rust, Corner the empress, Arxiv preprint arXiv:2204.11805 [math.CO], December 8 2022. Available at https://arxiv.org/abs/2204.11805.
47. Dominik Leon Jilg, Frobeniuszahl ausgewählter Zahlenfolgen: Analyse der Frobeniuszahl von synchronisierten Folgen und Folgen mit automatisierter charakteristischer Folge, Bachelor's thesis, Lehrstuhl für Mathematik IV, Julius-Maximilians-Universität Würzburg, Germany, August 12 2022.
48. Jeffrey Shallit, The Logical Approach To Automatic Sequences: Exploring Combinatorics on Words with Walnut, London Math. Soc. Lecture Note Series, Vol. 482, Cambridge University Press, September 29 2022.
49. Rob Burns, The appearance function for paper-folding words, arxiv preprint arXiv:2210.14719 [math.NT], October 22 2022.
50. Jeffrey Shallit, Some Tribonacci conjectures, arxiv preprint arXiv:2210.03996 [math.CO], October 8 2022. Here are the files for the automata you will need in this paper. Put them in the "Automaton Library" of Walnut.
    • xaut.txt
    • yaut.txt
51. Aseem Baranwal, James Currie, Lucas Mol, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Antisquares and critical exponents, arxiv preprint arXiv:2209.09223 [math.CO], September 19 2022.
52. Luke Schaeffer and Jeffrey Shallit, The First-Order Theory of Binary Overlap-Free Words is Decidable, arxiv preprint arXiv:2209.03266 [cs.FL], September 7 2022.
53. Jeffrey Shallit, Sonja Linghui Shan, and Kai Hsiang Yang, Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev, arxiv preprint arXiv:2208.06025 [cs.FL], August 11 2022, available at https://arxiv.org/abs/2208.06025. Appeared in J. Shallit, S. L. Shan and K. H. Yang, "Automatic sequences in negative bases and proofs of some conjectures of Shevelev", RAIRO-Theor. Inf. Appl. 57 (2023), 4.
54. Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, and Sonja Linghui Shan, Pseudoperiodic words and a question of Shevelev, Arxiv preprint, July 21 2022, http://arxiv.org/abs/2207.10171 . Final version appeared in Pseudoperiodic words and a question of Shevelev, Discrete Math. Theoret. Comput. Sci. 25(2) (2023).
55. Shuo Li, A note on the Lie complexity and beyond, Arxiv preprint arXiv:2207.05859 [math.CO], July 12 2022, available at https://arxiv.org/abs/2207.05859.
56. Golnaz Badkobeh, Tero Harju, Pascal Ochem, and Matthieu Rosenfeld, Avoiding square-free words on free groups, Theoretical Computer Science 922 (2022) 206-217.
57. James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a ternary infinite word. Arxiv preprint arXiv:2206.01776 [cs.DM], June 3 2022. Available at https://arxiv.org/abs/2206.01776. To reproduce the calculations, first download the latest version of Walnut here. Second, put the two files
    • msd_pisot4_addition.txt
    • msd_pisot4.txt
    in the Custom Bases directory. Third, put the file
    • PI.txt
    in the Word Automata directory. Fourth, put the files
    • pisi.txt
    • psynch0.txt
    • psynch1.txt
    • psynch2.txt
    in the Automata Library directory. Finally, all the commands from the paper are in the file
    • pis4.txt

58. R. Burns, Factorials and Legendre's three-square theorem: II, arxiv preprint arXiv:2203.16469 [math.NT], March 30 2022. Available at https://arxiv.org/abs/2203.16469.
59. J. Shallit, Note on a Fibonacci parity sequence, arxiv preprint arXiv:2203.10504 [cs.FL], March 20 2022. Available at https://arxiv.org/abs/2203.10504. Appeared in J. Shallit, Cryptography and Communications 15 (2023), 309-315.
60. N. Rampersad, The periodic complexity function of the Thue-Morse word, the Rudin-Shapiro word, and the period-doubling word, arxiv preprint arXiv:2112.04416, December 8 2021. Available at https://arxiv.org/abs/2112.04416. Appeared in Australasian J. Combinatorics 85 (2023), 150-158.
61. N. Rampersad, Prefixes of the Fibonacci word that end with a cube, arxiv preprint arXiv:2111.09253, November 17 2021. Available at https://arxiv.org/abs/2111.09253. Appeared as N. Rampersad, Prefixes of the Fibonacci word, C. R. Math. Acad. Sci. Paris 361 (2023) 323-330.
62. J. Shallit, Intertwining of Complementary Thue-Morse Factors, arxiv preprint arXiv:2203.02917 [cs.FL], March 6 2022. Available at https://arxiv.org/abs/2203.02917. Revised, published version in Australasian J. Combinatorics 84 (2022), 419-430. Available here .
63. J. Shallit, Sumsets of Wythoff sequences, Fibonacci representation, and beyond, Periodica Mathematica Hungarica 84 (2022), 37--46. Available here.
64. Phakhinkon Napp Phunphayap, Prapanpong Pongsriiam, and Jeffrey Shallit, Sumsets associated with Beatty sequences, to appear, Discrete Mathematics.
65. John Machacek, Mechanical proving with Walnut for squares and cubes in partial words, arxiv preprint arXiv:2201.05954 [cs.FL], January 16 2022. Appeared in CPM 2022, Leibniz Int'l Proc. Informatics, Vol. 223, 2022, pp.5:1-5:11.
66. J. Shallit, Additive Number Theory via Automata and Logic, arxiv preprint arXiv:2112.13627 [math.NT], December 27 2021.
67. G. Fici and J. Shallit, Properties of a Class of Toeplitz Words, arxiv preprint arXiv:2112.12125 [cs.FL], December 23 2021.
    • TP.txt file, place this in the Word Automata directory of Walnut
68. C. S. Kaplan and J. Shallit, A frameless 2-coloring of the plane lattice, Math. Magazine 94 (5) (2021), 353-360. A limited number of free-eprints are available here.
69. N. Rampersad and J. Shallit, Congruence properties of combinatorial sequences via Walnut and the Rowland-Yassawi-Zeilberger automaton, Arxiv preprint arXiv:2110.06244 [math.CO], October 12 2021. Appeared in Elect. J. Combinatorics 29 (3) (2022), P3.36.
70. J. Shallit, Synchronized sequences, in T. Lecroq and S. Puzynina, eds., WORDS 2021, LNICS 12847, Springer, 2021, pp. 1-19.
71. Jarkko Peltomäki and Ville Salo, Automatic winning shifts, Arxiv preprint arXiv:2106.07249 [cs.FL], June 14 2021.
72. J. Shallit, Hilbert's spacefilling curve described by automatic, regular, and synchronized sequences, Arxiv preprint arXiv:2106.01062 [cs.FL], June 2 2021. Files for the Walnut proofs:
    • HC.txt, put it in the Word Automata directory
    • Walnut commands for section 3
    • HS.txt, put it in the Word Automata directory
    • Walnut commands for section 5
    • Walnut commands for section 6
73. Jeffrey Shallit, Frobenius numbers and automatic sequences, Arxiv preprint arXiv:2103.10904 [math.NT], March 19 2021. Files for the paper:
    • complete Walnut code for all the proofs of the paper
    • shift.txt (put this in the "Automata Library" of Walnut)
    • fibinc.txt (put this in the "Automata Library" of Walnut)
74. Jason P. Bell and J. Shallit, Lie complexity of words, Arxiv preprint, Feb 7 2021, arXiv:2102.03821 [cs.FL]. Available here.
75. Aseem Baranwal, Luke Schaeffer, and Jeffrey Shallit, Ostrowski-automatic sequences: Theory and applications, Theor. Comput. Sci. 858 (2021) 122-142. Available here until March 11 2021.
76. Jeffrey Shallit, Robbins and Ardila meet Berstel, Info. Proc. Letters 167 (2021). Available at https://doi.org/10.1016/j.ipl.2020.106081.
77. Jeffrey Shallit, Abelian complexity and synchronization, arxiv preprint, November 1 2020. Appeared in Abelian complexity and synchronization, INTEGERS 21 (2021), #A36.
    Here are the files associated with the paper:
    • rst.txt (put this in the Result directory of Walnut)
    • TRL.txt (put this in the Word Automata Library)
    • Walnut commands for the paper
    • TRAC.txt (Walnut DFAO for abelian complexity of Tribonacci word; put in the Word Automata Library directory in order to use it)
    • TRAS.txt (Walnut DFAO for subsets of abelian factors of Tribonacci word; put in the Word Automata Library directory in order to use it)
78. Jeffrey Shallit, Subword complexity of the Fibonacci-Thue-Morse sequence: the proof of Dekking's conjecture, Indag. Math. 32 (2021), 729-735. Files for the paper:
    • FTM.txt, put this in the "Word Automata" library of Walnut
    • FTMD.txt, put this in the "Word Automata" library of Walnut
    • walnut commands
    • ftmatrix.mpl, the Maple code for the matrix

79. Jeffrey Shallit, Robbins and Ardila meet Berstel, Arxiv preprint 2007.14930, July 29 2020.
    • Walnut commands for the paper
80. Daniel Gabric and Jeffrey Shallit, The simplest binary word with only three squares, Arxiv preprint 2007.08188, July 17 2020. Data for the paper:
    • 3squares-proofs.txt
    • Q.txt
    • T1.txt
    • VTM.txt
    • GAB.txt
    • HN.txt
81. Marko Milosevic and Narad Rampersad, Squarefree words with interior disposable factors, Arxiv preprint, July 7 2020. Appeared in Theor. Comput. Sci. 863 (2021), 120-126.
82. Jarkko Peltomäki and Markus A. Whiteland, Avoiding abelian powers cyclically, Arxiv preprint, June 11 2020. Appeared in Advances in Applied Mathematics 121 (2020), 102095. DOI:10.1016/j.aam.2020.102095.
83. Aseem Raj Baranwal, Decision algorithms for Ostrowski-automatic sequences, Master's thesis, University of Waterloo, 2020.
84. Aseem Raj Baranwal, Jeffrey Shallit, Repetitions in infinite palindrome-rich words, arxiv preprint, April 22 2019. In Mercas R., Reidenbach D. (eds.) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science, vol. 11682, Springer, 2019, pp. 93-105. Available here.
85. Tim Ng, Pascal Ochem, Narad Rampersad, Jeffrey Shallit, New results on pseudosquare avoidance, arxiv preprint, April 19 2019. In Mercas R., Reidenbach D. (eds.) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science, vol. 11682, Springer, 2019, pp. 264-274. Available here.
86. T. Clokie, D. Gabric, and J. Shallit, Circularly squarefree words and unbordered conjugates: a new approach, arxiv preprint, April 17 2019. In Mercas R., Reidenbach D. (eds.) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science, vol. 11682, Springer, 2019, pp. 264-274. Available here.
87. Aseem R. Baranwal, Jeffrey Shallit, Critical exponent of infinite balanced words via the Pell number system, arxiv Preprint, February 1 2019. In Mercas R., Reidenbach D. (eds.) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science, vol. 11682, Springer, 2019, pp. 80-92. Available here.
88. James Currie, Narad Rampersad, Tero Harju, and Pascal Ochem, Some further results on squarefree arithmetic progressions in infinite words, arxiv preprint, January 18 2019. Appeared in Theoretical Computer Science 799 (2019) 140-148.
89. James Currie and Narad Rampersad, On some problems of Harju concerning squarefree arithmetic progressions in infinite words, arxiv preprint, December 5 2018.
90. Lukasz Merta, Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence, arxiv preprint, October 8 2018. Appeared in Discrete Mathematics and Theoretical Computer Science Vol. 22:1, 2020, #15.
91. Colin Krawchuk and Narad Rampersad, Cyclic Complexity of Some Infinite Words and Generalizations, Integers 18A (2018), #A12. Available at https://math.colgate.edu/~integers/sjs12/sjs12.pdf.
92. Jeffrey Shallit and Ramin Zarifi, Circular critical exponents for Thue–Morse factors, RAIRO Info. Theor., published online 17 January 2019. Available here.
    • Walnut commands for the Thue-Morse sequence results
    • Walnut commands for the paperfolding sequence
93. Pierre Bonardo, Anna E. Frid, Jeffrey Shallit, "The number of valid factorizations of Fibonacci prefixes", Theor. Comput. Sci., 2019, to appear. Available online at https://doi.org/10.1016/j.tcs.2018.12.016.
94. Jason Bell, Kathryn Hare and Jeffrey Shallit, When is an automatic set an additive basis? Proc. Amer. Math. Soc. Ser. B 5 (2018), 50-63. Available here.
    • examples
95. Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arxiv preprint, April 23 2018. Appeared in M. Hoshi and S. Seki, eds., DLT 2018, LNCS Vol. 11088, Springer, 2018, pp. 121-132. Here are text files describing how you can verify the results yourself using Grail and Walnut.
    • Theorem 2
    • Theorem 5
    • Theorem 7
    • Theorem 9
    • Theorem 12
    • Theorem 15
96. Narad Rampersad, Jeffrey Shallit, Élise Vandomme, Critical exponents of infinite balanced words, arxiv preprint, January 16 2018. Accepted for Theoretical Computer Science; in press here.
97. James Currie, Lucas Mol, and Narad Rampersad, A family of formulas with reversal of high avoidability index, International Journal of Algebra and Computation 27 (2017) 477-493.
98. A. Rajasekaran, N. Rampersad, J. Shallit, Overpals, Underlaps, and Underpals, in WORDS 2017: Combinatorics on Words, 2017, pp. 17-29.
99. Chen Fei Du, Hamoon Mousavi, Eric Rowland, Luke Schaeffer, and Jeffrey Shallit, Decision Algorithms for Fibonacci-Automatic Words, II: Related Sequences and Avoidability, Theoret. Comput. Sci. 657 (2017), 146-162. Examples from the paper that can be run with Walnut (download Walnut below):
    • Thue-Morse examples
    • Rudin-Shapiro examples
    • Rudin-Shapiro example for trapezoidal words
    • paperfolding sequence examples
    • paperfolding example for trapezoidal words
    • period-doubling examples
    • Fibonacci examples
    You'll also need to put the following files in the "Word Automata" library:
    • RS.txt (automaton for Rudin-Shapiro)
    • P.txt (msd automaton for paperfolding)
    • PR.txt (lsd automaton for paperfolding)
    • PD.txt (automaton for period-doubling)

100. Quentin Valembois, Propriétés décidables des suites automatiques, Master's thesis, University of Liège, Belgium, 2017. Available here.
101. Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Elect. J. Combinatorics 23 (1) (2016), Paper #P1.25.
102. Chen Fei Du, Hamoon Mousavi, Luke Schaeffer, and Jeffrey Shallit, Decision Algorithms for Fibonacci-Automatic Words, III: Enumeration and Abelian Properties, Int. J. Found. Comput. Sci. 27 (8) (2016), 943-963.
103. Hamoon Mousavi, Luke Schaeffer, and Jeffrey Shallit, Decision Algorithms for Fibonacci-Automatic Words, I: Basic Results, RAIRO Inform. Théorique 50 (2016), 39-66. E-version: here.
104. Hamoon Mousavi, Automatic Theorem Proving in Walnut, Arxiv preprint arXiv:1603.06017v1 [cs.FL], March 18 2016. Revised version available at arXiv:1603.06017v2 [cs.FL], May 25 2021.
105. Floriane Magera, Automatic demonstration of mathematical conjectures, Master's thesis, University of Liège, 2015-6. Available at https://matheo.uliege.be/handle/2268.2/1679.
106. Adam Borchert and Narad Rampersad, Words with many palindrome pair factors, Arxiv preprint arXiv:1509.05396 [math.CO], September 17 2015. Published version in Electronic J. Combinatorics 22 (4) (2015), Paper P4.23.
107. Daniel Goc, Hamoon Mousavi, Luke Schaeffer, Jeffrey Shallit, A New Approach to the Paperfolding Sequences, in Arnold Beckmann, Victor Mitrana, Mariya Soskova, eds., Evolving Computability: 11th Conference on Computability in Europe, CiE 2015, Springer, LNICS, Vol. 9136, 2015, pp. 34-43. Available here.
108. H. Mousavi and J. Shallit, Mechanical Proofs of Properties of the Tribonacci Word, in F. Manea and D. Nowotka, eds., WORDS 2015, LNCS 9304, Springer, 2015, pp. 1-21. Here are the Walnut predicates you can use to reproduce most of the results in this paper:
     • predicates
     • TR.txt, put in the "Word Automata" library
     This was one of the first Walnut papers we wrote, and now (in 2024) we know there are better ways to do some of the things we did in this one. For example, to check whether TR[i..i+n-1] is a palindrome, where TR denotes the Tribonacci word, one can simply write
     def tribpal "?msd_trib Au,v (u>=i & u<i+n & u+v+1=2*i+n) => TR[u]=TR[v]":
     and hence avoid the split into even and odd cases. It runs very quickly!

     Similarly, to check that TR is mirror invariant, one can write
     def trib_fac_reverse_equiv "?msd_trib Au,v (u>=i & u<i+n & u+v+1=i+j+n) => TR[u]=TR[v]":
     def trib_mirror "?msd_trib Ai,n Ej $trib_fac_reverse_equiv(i,j,n)":
     which also runs rather quickly.

109. Daniel Goc, Narad Rampersad, Michel Rigo, Pavel Salimov, On the number of Abelian Bordered Words (with an Example of Automatic Theorem-Proving) Int. J. Found. Comput. Sci. 25 (2014), 1097-1110.