@article{Onn1999,
title = {Cutting Corners},
journal = {Advances in Applied Mathematics},
volume = {23},
number = {1},
pages = {29-48},
year = {1999},
issn = {0196-8858},
doi = {https://doi.org/10.1006/aama.1999.0645},
url = {https://www.sciencedirect.com/science/article/pii/S0196885899906450},
author = {Shmuel Onn and Bernd Sturmfels}
}

@article{Andrews1963,
 ISSN = {00029947},
 URL = {http://www.jstor.org/stable/1993769},
 author = {George E. Andrews},
 journal = {Transactions of the American Mathematical Society},
 number = {2},
 pages = {270--279},
 publisher = {American Mathematical Society},
 title = {A Lower Bound for the Volume of Strictly Convex Bodies with many Boundary Lattice Points},
 urldate = {2023-05-07},
 volume = {106},
 year = {1963}
}

@article{Corteel1999,
title = {The Number of Plane Corner Cuts},
journal = {Advances in Applied Mathematics},
volume = {23},
number = {1},
pages = {49-53},
year = {1999},
issn = {0196-8858},
doi = {https://doi.org/10.1006/aama.1999.0646},
url = {https://www.sciencedirect.com/science/article/pii/S0196885899906462},
author = {Sylvie Corteel and Gaël Rémond and Gilles Schaeffer and Hugh Thomas}
}

@article{Bergeron2023,
 AUTHOR = {Bergeron, Fran\c{c}ois and Mazin, Mikhail},
     TITLE = {Combinatorics of triangular partitions},
   JOURNAL = {Enumer. Comb. Appl.},
  FJOURNAL = {Enumerative Combinatorics and Applications},
    VOLUME = {3},
      YEAR = {2023},
    NUMBER = {1},
     PAGES = {Paper No. S2R1, 20},
   MRCLASS = {05A17 (05E05 05E10)},
  MRNUMBER = {4528166},
       DOI = {10.54550/eca2023v3s1r1},
       URL = {https://doi.org/10.54550/eca2023v3s1r1},
}

@article{Boshernitzan1981,
title = {Nonhomogeneous spectra of numbers},
journal = {Discrete Mathematics},
volume = {34},
number = {3},
pages = {325-327},
year = {1981},
issn = {0012-365X},
doi = {https://doi.org/10.1016/0012-365X(81)90013-3},
url = {https://www.sciencedirect.com/science/article/pii/0012365X81900133},
author = {M. Boshernitzan and A.S. Fraenkel},
abstract = {A simple characterization is given of those sequences of integersMn={ai}ni=1for which there exist real numbers αandβ such thatai=⌊iα+β⌋(1⩽i⩽n). In addition, for givenMn, an open intervalInis computed such that αϵInif and only ifai=⌊iα+β⌋(1⩽i⩽n)for suitableβ=β(α).}
}

@INPROCEEDINGS{Bruckstein1990,
  author={Bruckstein, A.M.},
  booktitle={[1990] Proceedings. 10th International Conference on Pattern Recognition}, 
  title={The self-similarity of digital straight lines}, 
  year={1990},
  volume={i},
  number={},
  pages={485-490 vol.1},
  doi={10.1109/ICPR.1990.118151}}

  @article{Wagner2002,
title = {On the number of corner cuts},
journal = {Advances in Applied Mathematics},
volume = {29},
number = {2},
pages = {152-161},
year = {2002},
issn = {0196-8858},
doi = {https://doi.org/10.1016/S0196-8858(02)00014-3},
url = {https://www.sciencedirect.com/science/article/pii/S0196885802000143},
author = {Uli Wagner},
keywords = {Corner cut, Gröbner fan, -set, Reverse search}
}

@inbook{Lothaire2002, place={Cambridge}, series={Encyclopedia of Mathematics and its Applications}, title={Sturmian Words}, DOI={10.1017/CBO9781107326019.003}, booktitle={Algebraic Combinatorics on Words}, publisher={Cambridge University Press}, author={Lothaire, M.}, year={2002}, pages={45–110}, collection={Encyclopedia of Mathematics and its Applications}}

@Article{Lipatov1982,
 Author = {Lipatov, E. P.},
 Title = {On a classification of binary collections and properties of homogeneity classes},
 FJournal = {Problemy Kibernetiki},
 Journal = {Probl. Kibern.},
 Volume = {39},
 Pages = {67--84},
 Year = {1982},
 Language = {Russian},
 Keywords = {94A99,94C10,05A15},
 zbMATH = {3885787},
 Zbl = {0555.94007}
}

@article{Mignosi1991,
title = {On the number of factors of Sturmian words},
journal = {Theoretical Computer Science},
volume = {82},
number = {1},
pages = {71-84},
year = {1991},
issn = {0304-3975},
doi = {https://doi.org/10.1016/0304-3975(91)90172-X},
url = {https://www.sciencedirect.com/science/article/pii/030439759190172X},
author = {Filippo Mignosi},
abstract = {We prove that for m⩾1, card(Am) = 1+∑mi=1 (m−i+1)ϕ(i) where Am is the set of factors of length m of all the Sturmian words and ϕ is the Euler function. This result was conjectured by Dulucq and Gouyou-Beauchamps (1987) who proved that this result implies that the language (∪m⩾0Am)c is inherently ambiguous. We also give a combinatorial version of the Riemann hypothesis.}
}

@article{Berstel93,
AUTHOR = {Berstel, Jean and Pocchiola, Michel},
     TITLE = {A geometric proof of the enumeration formula for {S}turmian
              words},
   JOURNAL = {Internat. J. Algebra Comput.},
  FJOURNAL = {International Journal of Algebra and Computation},
    VOLUME = {3},
      YEAR = {1993},
    NUMBER = {3},
     PAGES = {349--355},
      ISSN = {0218-1967},
   MRCLASS = {05A15 (68R15)},
  MRNUMBER = {1240390},
MRREVIEWER = {Giuseppe Pellegrino},
       DOI = {10.1142/S0218196793000238},
       URL = {https://doi.org/10.1142/S0218196793000238},
}

@misc{Cassaigne2001,
      title={Complexity and growth for polygonal billiards}, 
      author={J. Cassaigne and P. Hubert and S. Troubetzkoy},
      year={2001},
      eprint={math/0109208},
      archivePrefix={arXiv},
      primaryClass={math.DS}
}

@ARTICLE{Berenstein88,
  author={Berenstein, C.A. and Lavine, D.},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, 
  title={On the number of digital straight line segments}, 
  year={1988},
  volume={10},
  number={6},
  pages={880-887},
  doi={10.1109/34.9109}}

  @article{Akiyama2022,
title = {Asymptotic formula for balanced words},
journal = {Journal of Number Theory},
volume = {236},
pages = {144-159},
year = {2022},
issn = {0022-314X},
doi = {https://doi.org/10.1016/j.jnt.2021.07.014},
url = {https://www.sciencedirect.com/science/article/pii/S0022314X21002699},
author = {Shigeki Akiyama},
keywords = {Irrational rotation, Balanced words, Sturmian words, Farey fraction, Riemann hypothesis, Large sieve},
abstract = {We give asymptotic formulas for the number of balanced words whose slope α and intercept ρ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case, the error term is deduced using an inequality of large sieve type.}
}

@misc{oeis,
    Author = {{OEIS Foundation Inc.}},
    Note = {Published electronically at \url{http://oeis.org}},
    Title = {The {O}n-{L}ine {E}ncyclopedia of {I}nteger {S}equences},
    Year = 2023}

@article {Blasiak2023,
    AUTHOR = {Blasiak, Jonah and Haiman, Mark and Morse, Jennifer and Pun,
              Anna and Seelinger, George H.},
     TITLE = {A shuffle theorem for paths under any line},
   JOURNAL = {Forum Math. Pi},
  FJOURNAL = {Forum of Mathematics. Pi},
    VOLUME = {11},
      YEAR = {2023},
     PAGES = {Paper No. e5, 38},
   MRCLASS = {05A15 (20G05 35P15)},
  MRNUMBER = {4553914},
       DOI = {10.1017/fmp.2023.4},
       URL = {https://doi.org/10.1017/fmp.2023.4},
}

@misc{website,
    note= {Code available at \url{https://math.dartmouth.edu/~sergi/tp}}
}