Combinatorial Properties of Fibonacci Arrays

  • Manasi S. Kulkarni
  • Kalpana MahalingamEmail author
  • Sivasankar Mohankumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)


The non-trivial extension of Fibonacci words to Fibonacci arrays was proposed by Apostolico and Brimkov in order to study repetitions in arrays. In this paper we investigate several combinatorial as well as formal language theoretic properties of Fibonacci arrays. In particular, we show that the set of all Fibonacci arrays is a 2D primitive language (under certain conditions), count the number of borders in Fibonacci arrays, and show that the set of all Fibonacci arrays is a non-recognizable language. We also show that the set of all square Fibonacci arrays is a two dimensional code.


Fibonacci words Fibonacci arrays Recognizable picture language Two dimensional code Primitivity 


  1. 1.
    Alfred, S.P., Ingmar, L.: The Fabulous Fibonacci Numbers. Prometheus Books (2007)Google Scholar
  2. 2.
    Ali, D.: Twelve simple algorithms to compute Fibonacci numbers. CoRR. (2018)
  3. 3.
    Amir, A., Benson, G.: Two-dimensional periodicity in rectangular arrays. SIAM J. Comput. 27(1), 90–106 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Anselmo, M., Giammarresi, D., Madonia, M.: Two dimensional prefix codes of pictures. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 46–57. Springer, Heidelberg (2013). Scholar
  5. 5.
    Anselmo, M., Giammarresi, D., Madonia, M.: Prefix picture codes: a decidable class of two-dimensional codes. Int. J. Found. Comput. Sci. 25(8), 1017–1031 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Anselmo, M., Madonia, M.: Two-dimensional comma-free and cylindric codes. Theor. Comput. Sci. 658, 4–17 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Apostolico, A., Brimkov, V.E.: Fibonacci arrays and their two-dimensional repetitions. Theor. Comput. Sci. 237(1–2), 263–273 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berthé, V., Vuillon, L.: Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discret. Math. 223(1–3), 27–53 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Berthé, V., Vuillon, L.: Palindromes and two-dimensional Sturmian sequences. J. Autom. Lang. Comb. 6(2), 121–138 (2001)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bozapalidis, S., Grammatikopoulou, A.: Picture codes. RAIRO-Theor. Inform. Appl. 40(4), 537–550 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Carlitz, L.: Fibonacci representations II. Fibonacci Q. 8, 113–134 (1970)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chang, C., Wang, H.: Comparison of two-dimensional string matching algorithms. In: 2012 International Conference on Computer Science and Electronics Engineering, vol. 3, pp. 608–611. IEEE (2012)Google Scholar
  13. 13.
    Dallapiccola, R., Gopinath, A., Stellacci, F., Dal, N.L.: Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles. Opt. Express 16(8), 5544–5555 (2008)CrossRefGoogle Scholar
  14. 14.
    Fu, K.S.: Syntactic Methods in Pattern Recognition, vol. 112. Elsevier, Amsterdam (1974)zbMATHGoogle Scholar
  15. 15.
    Gamard, G., Richomme, G., Shallit, J., Smith, T.J.: Periodicity in rectangular arrays. Inf. Process. Lett. 118, 58–63 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Geizhals, S., Sokol, D.: Finding maximal 2-dimensional palindromes. In: The Proceedings of the 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, vol. 54, no. 19, pp. 1–12. Dagstuhl (2016)Google Scholar
  17. 17.
    Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 215–267. Springer, Heidelberg (1997). Scholar
  18. 18.
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation, vol. 24. Pearson (2006)Google Scholar
  19. 19.
    Kulkarni, M.S., Mahalingam, K.: Two-dimensional palindromes and their properties. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 155–167. Springer, Cham (2017). Scholar
  20. 20.
    Lothaire, M.: Algebraic Combinatorics on Words, vol. 90. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  21. 21.
    Luca, A.: A combinatorial property of the Fibonacci words. Inf. Process. Lett. 12(4), 193–195 (1981)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Margenstern, M.: An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl. Math. Comput. 218(16), 8005–8018 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Margenstern, M.: Fibonacci words, hyperbolic tilings and grossone. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 3–11 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mignosi, F., Restivo, A., Silva, P.V.: On Fine and Wilf’s theorem for bidimensional words. Theor. Comput. Sci. 292(1), 245–262 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Minsky, M., Papert, S.: Perceptrons. The MIT Press, Cambridge (1969)zbMATHGoogle Scholar
  26. 26.
    Pal, D., Masami, I.: Primitive words and palindromes. In: Context-Free Languages and Primitive Words, pp. 423–435. World Scientific (2014)Google Scholar
  27. 27.
    Yu, S.S., Zhao, Y.K.: Properties of Fibonacci languages. Discret. Math. 224(1–3), 215–223 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

Personalised recommendations