IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v33y2007i1p38-49.html
   My bibliography  Save this article

The k-Fibonacci sequence and the Pascal 2-triangle

Author

Listed:
  • Falcón, Sergio
  • Plaza, Ángel

Abstract

The general k-Fibonacci sequence {Fk,n}n=0∞ were found by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition. This sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the so-called Pascal 2-triangle.

Suggested Citation

  • Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.
    • Handle: RePEc:eee:chsofr:v:33:y:2007:i:1:p:38-49
    DOI: 10.1016/j.chaos.2006.10.022
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S096007790600974X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2006.10.022?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. El Naschie, M.S., 2007. "The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 537-547.
    2. El Naschie, M.S., 2005. "Anomaly cancellation and the mass spectrum of ε(∞)," Chaos, Solitons & Fractals, Elsevier, vol. 23(3), pages 1087-1090.
    3. Stakhov, Alexey & Rozin, Boris, 2006. "The continuous functions for the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1014-1025.
    4. Stakhov, A.P., 2006. "Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 56-66.
    5. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    6. El Naschie, M.S., 2006. "Elementary prerequisites for E-infinity," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 579-605.
    7. El Naschie, M.S., 2005. "Non-Euclidean spacetime structure and the two-slit experiment," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 1-6.
    8. El Naschie, M.S., 2006. "Topics in the mathematical physics of E-infinity theory," Chaos, Solitons & Fractals, Elsevier, vol. 30(3), pages 656-663.
    9. El Naschie, M.S., 2005. "Dead or alive: Desperately seeking Schrödinger’s cat," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 673-676.
    10. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
    11. Naschie, M.S.El, 2005. "Deriving the essential features of the standard model from the general theory of relativity," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 941-946.
    12. Stakhov, Alexey, 2006. "Fundamentals of a new kind of mathematics based on the Golden Section," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1124-1146.
    13. Stakhov, A. & Rozin, B., 2006. "The “golden” algebraic equations," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1415-1421.
    14. El Naschie, M.S., 2005. "Stability Analysis of the two-slit experiment with quantum particles," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 291-294.
    15. El Naschie, M.S., 2007. "Hilbert space, Poincaré dodecahedron and golden mean transfiniteness," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 787-793.
    16. Stakhov, A.P., 2005. "The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering," Chaos, Solitons & Fractals, Elsevier, vol. 26(2), pages 263-289.
    17. El Naschie, M.S., 2005. "Experimental and theoretical arguments for the number and the mass of the Higgs particles," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1091-1098.
    18. El Naschie, M.S., 2005. "On the cohomology and instantons number in E-infinity Cantorian spacetime," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 13-17.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Aydın, Fügen Torunbalcı, 2018. "Dual-complex k-Fibonacci numbers," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 1-6.
    2. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    3. Bijendra Singh & Kiran Sisodiya & Farooq Ahmad, 2014. "On the Products of -Fibonacci Numbers and -Lucas Numbers," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-4, June.
    4. Pavel Trojovský & Štěpán Hubálovský, 2020. "Some Diophantine Problems Related to k -Fibonacci Numbers," Mathematics, MDPI, vol. 8(7), pages 1-10, June.
    5. Falcón, Sergio & Plaza, Ángel, 2008. "The k-Fibonacci hyperbolic functions," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 409-420.
    6. Flaut, Cristina & Savin, Diana, 2019. "Some remarks regarding l-elements defined in algebras obtained by the Cayley–Dickson process," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 112-116.
    7. Can Kızılateş & Tiekoro Kone, 2023. "On special spacelike hybrid numbers with Fibonacci divisor number components," Indian Journal of Pure and Applied Mathematics, Springer, vol. 54(1), pages 279-287, March.
    8. Esmaeili, M. & Gulliver, T.A. & Kakhbod, A., 2009. "The Golden mean, Fibonacci matrices and partial weakly super-increasing sources," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 435-440.
    9. Falcon, Sergio & Plaza, Ángel, 2009. "k-Fibonacci sequences modulo m," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 497-504.
    10. Flaut, Cristina & Savin, Diana, 2018. "Some special number sequences obtained from a difference equation of degree three," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 67-71.
    11. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.
    12. Falcon, Sergio, 2016. "The k–Fibonacci difference sequences," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 153-157.
    13. Esmaeili, Morteza & Esmaeili, Mostafa, 2009. "Polynomial Fibonacci–Hessenberg matrices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2820-2827.
    14. Natalia Bednarz, 2021. "On ( k , p )-Fibonacci Numbers," Mathematics, MDPI, vol. 9(7), pages 1-13, March.
    15. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    16. Akbulak, Mehmet & Bozkurt, Durmuş, 2009. "On the order-m generalized Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1347-1355.
    17. Falcón, Sergio & Plaza, Ángel, 2008. "On the 3-dimensional k-Fibonacci spirals," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 993-1003.
    18. Flaut, Cristina & Shpakivskyi, Vitalii & Vlad, Elena, 2017. "Some remarks regarding h(x) – Fibonacci polynomials in an arbitrary algebra," Chaos, Solitons & Fractals, Elsevier, vol. 99(C), pages 32-35.
    19. Flavio Pressacco & Giacomo Plazzotta & Laura Ziani, 2014. "K-Fibonacci sequences and minimal winning quota in Parsimonious game," Working Papers hal-00950090, HAL.
    20. Ghorbani, Modjtaba & Hakimi-Nezhaad, Mardjan & Dehmer, Matthias, 2022. "Novel results on partial hosoya polynomials: An application in chemistry," Applied Mathematics and Computation, Elsevier, vol. 433(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Falcón, Sergio & Plaza, Ángel, 2007. "On the Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1615-1624.
    2. Falcón, Sergio & Plaza, Ángel, 2008. "On the 3-dimensional k-Fibonacci spirals," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 993-1003.
    3. Falcón, Sergio & Plaza, Ángel, 2008. "The k-Fibonacci hyperbolic functions," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 409-420.
    4. Falcón, Sergio & Plaza, Ángel, 2009. "On k-Fibonacci sequences and polynomials and their derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1005-1019.
    5. Kocer, E. Gokcen & Tuglu, Naim & Stakhov, Alexey, 2009. "On the m-extension of the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1890-1906.
    6. Akbulak, Mehmet & Bozkurt, Durmuş, 2009. "On the order-m generalized Fibonacci k-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1347-1355.
    7. Falcón, Sergio & Plaza, Ángel, 2009. "The metallic ratios as limits of complex valued transformations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 1-13.
    8. Falcon, Sergio & Plaza, Ángel, 2009. "k-Fibonacci sequences modulo m," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 497-504.
    9. Ekici, Erdal & Noiri, Takashi, 2009. "Decompositions of continuity, α-continuity and AB-continuity," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2055-2061.
    10. Ekici, Erdal, 2009. "A note on almost β-continuous functions," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 1010-1013.
    11. Kılıç, Emrah, 2009. "The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 2047-2063.
    12. Esmaeili, Morteza & Esmaeili, Mostafa, 2009. "Polynomial Fibonacci–Hessenberg matrices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2820-2827.
    13. Ekici, Erdal, 2008. "Generalization of weakly clopen and strongly θ-b-continuous functions," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 79-88.
    14. El Naschie, M.S., 2007. "On the universality class of all universality classes and E-infinity spacetime physics," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 927-936.
    15. Stakhov, Alexey, 2007. "The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 315-334.
    16. Büyükkılıç, F. & Demirhan, D., 2009. "Cumulative growth with fibonacci approach, golden section and physics," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 24-32.
    17. Stakhov, A.P., 2007. "The “golden” matrices and a new kind of cryptography," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1138-1146.
    18. El Naschie, Mohamed Saladin, 2006. "The idealized quantum two-slit gedanken experiment revisited—Criticism and reinterpretation," Chaos, Solitons & Fractals, Elsevier, vol. 27(4), pages 843-849.
    19. El Naschie, M.S., 2007. "Estimating the experimental value of the electromagnetic fine structure constant α¯0=1/137.036 using the Leech lattice in conjunction with the monster group and Spher’s kissing number in 24 dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 383-387.
    20. Stakhov, Alexey, 2006. "The golden section, secrets of the Egyptian civilization and harmony mathematics," Chaos, Solitons & Fractals, Elsevier, vol. 30(2), pages 490-505.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:33:y:2007:i:1:p:38-49. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.