References

  1. Aghigh K., Masjed-Jamei M., Dehghan M. A survey on third and fourth kind of Chebyshev polynomials and their applications. Appl. Math. Comput. 2008, 199 (1), 2–12. doi:10.1016/j.amc.2007.09.018
  2. Aigner M. Catalan and other numbers: a recurrent theme. Springer, Milan, 2001.
  3. Balaich M., Ondrus M. A generalization of even and odd functions. Involve J. Math. 2011, 4 (1), 91–102. doi:10.2140/involve.2011.4.91
  4. Barbeau E. Pell’s equation. Problem books in mathematics. Springer, New York, 2003.
  5. Beauregard R.A., Dobrushkin V.A. Multisection of series. Math. Gaz. 2016, 100 (549), 460–470. doi:10.1017/mag.2016.111
  6. Bertrand J. Solution d’un problème. C. R. Acad. Sci. Sér. I (Math.) 1887, 105, 369.
  7. Bollinger R.C. Extended Pascal triangles. Math. Mag. 1993, 66 (2), 87–94. doi:10.2307/2691114
  8. Bondarenko B.A. Generalized Pascal triangles and pyramids, their fractals, graphs and applications. The Fibonacci Association, Santa Clara, 1993.
  9. Chow T., West J. Forbidden subsequences and Chebyshev polynomials. Discrete Math. 1999, 204 (1), 119–128. doi:10.1016/S0012-365X(98)00384-7
  10. Cigler J. Some remarks and conjectures related to lattice paths in strips along the \(x\)-axis. arXiv:1501.04750 [math.CO] doi:10.48550/arxiv.1501.04750
  11. de Bruijn N.G., Knuth D.E., Rice S.O. The average height of planted plane trees. Graph Theory & Comput. 1972, 1972, 15–22. doi:10.1016/B978-1-4832-3187-7.50007-6
  12. Deng L.H., Deng Y.P., Shapiro L.W. The Riordan group and symmetric lattice paths. J. Shandong Univ. Nat. Sci. 2015, 50 (4), 82–89. doi:10.6040/j.issn.1671-9352.0.2014.196
  13. Dershowitz N. Between Broadway and the Hudson: A bijection of corridor paths. J. Integer Seq. 2021, 24 (2), article 21.2.8.
  14. Dubeau F. Newton’s method and high-order algorithms for the \(n\)th root computation. J. Comput. Appl. Math. 2009, 224 (1), 66–76. doi:10.1016/j.cam.2008.04.014
  15. Ferrari L. Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations. Graphs Combin. 2010, 26 (1), 51–70. doi:10.1007/s00373-010-0895-z
  16. Fowler D., Robson E. Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context. Hist. Math. 1998, 25 (4), 366–378. doi:10.1006/hmat.1998.2209
  17. Guibert O., Mansour T. Restricted 132-involutions. Sémin. Lothar. Comb. 2002, 48, B48a.
  18. Hasan M.A. New families of higher order iterative methods for solving equations. In: Proc. of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, December 13–15, 2006. IEEE, 2007, 6379–6384.
  19. Hein N., Huang J. Variations of the Catalan numbers from some nonassociative binary operations. Discrete Math. 2022, 345 (3), 112711. doi:10.1016/j.disc.2021.112711
  20. Householder A. The numerical treatment of a single nonlinear equation. McGraw-Hill, New-York, 1970.
  21. Kosheleva O. Babylonian method of computing the square root: justifications based on fuzzy techniques and on computational complexity. In: NAFIPS 2009 – 2009 Annual Meeting of the North American Fuzzy Information Processing Society, Cincinnati, OH, USA , June 14–17, 2009. IEEE, 2009, 1–6.
  22. Krattenthaler C. Permutations with restricted patterns and Dyck paths. Adv. Appl. Math. 2001, 27 (2–3), 510–530. doi:10.1006/aama.2001.0747
  23. Kreweras G. Sur les éventails de segments. Cah. Bur. Univ. Rech. Opérationnelle 1970, 15, 3–41.
  24. Lima J.B., Campello de Souza R.M. Tangent function and Chebyshev-like rational maps over finite fields. IEEE Trans. Circuits Syst. II, Exp. Briefs 2020, 67 (4), 775–779. doi:10.1109/TCSII.2019.2923879
  25. Lipton R.J., Zalcstein Y. Word problems solvable in logspace. J. ACM 1977, 24 (3), 522–526. doi:10.1145/322017.322031
  26. Mason J. Handscomb D. Chebyshev polynomials. CRC Press LLC, Boca Raton, 2002.
  27. McBride A. Remarks on Pell’s equation and square root algorithms. Math. Gaz. 1999, 83 (496), 47–52. doi:10.2307/3618682
  28. Polyak B. Newton’s method and its use in optimization. Eur. J. Oper. Res. 2007, 181 (3), 1086–1096. doi:10.1016/j.ejor.2005.06.076
  29. Ricci P.E. A survey on pseudo-Chebyshev functions. 4open 2020, 3, 2. doi:10.1051/fopen/2020001
  30. Sloane N.J.A. The on-line encyclopedia of integer sequences (OEIS). https://oeis.org/
  31. Yeyios A.K. On two sequences of algorithms for approximating square roots. J. Comput. Appl. Math. 1992, 40 (1), 63–72. doi:10.1016/0377-0427(92)90042-V