IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v192y2022icp278-290.html
   My bibliography  Save this article

Generalized Pascal’s triangles and associated k-Padovan-like sequences

Author

Listed:
  • Anatriello, Giuseppina
  • Németh, László
  • Vincenzi, Giovanni

Abstract

One of the most interesting properties of Pascal’s triangle is that the sequence of the sums of the elements on its diagonals is the best known recurrence sequence, the Fibonacci sequence. It is also known that other diagonals can be associated with other relevant recurrence sequences, such as the Padovan and k-Padovan sequences. In this paper, we see that similar properties also hold for diagonals of generalized Pascal’s triangles. We show that the diagonal sums in generalized Pascal’s triangles belong to the family of the so-called ‘k-Padovan-like sequences’ which are linear recurrences of order k with constant coefficients. A recurrence connection between the k-Padovan and k-Padovan-like sequences is derived.

Suggested Citation

  • Anatriello, Giuseppina & Németh, László & Vincenzi, Giovanni, 2022. "Generalized Pascal’s triangles and associated k-Padovan-like sequences," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 192(C), pages 278-290.
    • Handle: RePEc:eee:matcom:v:192:y:2022:i:c:p:278-290
    DOI: 10.1016/j.matcom.2021.09.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421003219
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.09.006?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. Belbachir, Hacène & Németh, László & Szalay, László, 2016. "Hyperbolic Pascal triangles," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 453-464.
    2. Gogin, N. & Mylläri, A., 2016. "Padovan-like sequences and Bell polynomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 125(C), pages 168-177.
    3. Nazmiye Yilmaz & Necati Taskara, 2013. "Binomial Transforms of the Padovan and Perrin Matrix Sequences," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, October.
    4. Alberto Fiorenza & Giovanni Vincenzi, 2013. "From Fibonacci Sequence to the Golden Ratio," Journal of Mathematics, Hindawi, vol. 2013, pages 1-3, March.
    5. Halici, Serpil & Karataş, Adnan, 2017. "On a generalization for fibonacci quaternions," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 178-182.
    6. Fiorenza, Alberto & Vincenzi, Giovanni, 2011. "Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 145-152.
    7. 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.
    Full references (including those not matched with items on IDEAS)

    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. Ivana Matoušová & Pavel Trojovský, 2020. "On Coding by (2, q )-Distance Fibonacci Numbers," Mathematics, MDPI, vol. 8(11), pages 1-24, November.
    2. Renato Fiorenza, 2022. "Existence of the Limit of Ratios of Consecutive Terms for a Class of Linear Recurrences," Mathematics, MDPI, vol. 10(12), pages 1-8, June.
    3. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.
    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. E. Gokcen Kocer & Huriye Alsan, 2022. "Generalized Hybrid Fibonacci and Lucas p-numbers," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(4), pages 948-955, December.
    6. Esmaeili, Morteza & Esmaeili, Mostafa, 2009. "Polynomial Fibonacci–Hessenberg matrices," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2820-2827.
    7. Stakhov, A.P., 2007. "The “golden” matrices and a new kind of cryptography," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1138-1146.
    8. Alberto Fiorenza & Giovanni Vincenzi, 2013. "From Fibonacci Sequence to the Golden Ratio," Journal of Mathematics, Hindawi, vol. 2013, pages 1-3, March.
    9. Falcón, Sergio & Plaza, Ángel, 2007. "The k-Fibonacci sequence and the Pascal 2-triangle," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 38-49.
    10. 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.
    11. Fiorenza, Alberto & Vincenzi, Giovanni, 2011. "Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 44(1), pages 145-152.
    12. Flavio Pressacco & Giacomo Plazzotta & Laura Ziani, 2014. "K-Fibonacci sequences and minimal winning quota in Parsimonious game," Working Papers hal-00950090, HAL.
    13. Ayşe Zeynep Azak, 2022. "Pauli Gaussian Fibonacci and Pauli Gaussian Lucas Quaternions," Mathematics, MDPI, vol. 10(24), pages 1-10, December.
    14. 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.
    15. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    16. Kilic, E. & Stakhov, A.P., 2009. "On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2210-2221.
    17. Deveci, Ömür & Hulku, Sakine & Shannon, Anthony G., 2021. "On the co-complex-type k-Fibonacci numbers," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).

    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:matcom:v:192:y:2022:i:c:p:278-290. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    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.