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Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 3

An extension of Lucas identity via Pascal's triangle

Giuseppina ANATRIELLO
Giovanni VINCENZI
Pages
647–658
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity.
📊
Key Findings & Data:An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences.
💡
Conclusion & Impact:A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

The Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity. An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences. This family both contains the Fibonacci and the Padovan sequences. A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.

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Cite This Article

ANATRIELLO, G., VINCENZI, G. (2022). An extension of Lucas identity via Pascal's triangle. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 3, 647–658. https://ajindex.com/articles/529545

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References (18)

  1. Z. Akyuz and S. Halici, On some combinatorial identities involving the terms of generalized Fibonacci and Lucas sequences, Hacet. J. Math. Stat. 42 (4), 431–435, 2013
  2. G. Anatriello and G. Vincenzi, Tribonacci-like sequences and generalized Pascal’s triangles, Internat. J. Math. Ed. Sci. Tech. 45 (8), 1220–1232, 2014
  3. G. Anatriello and G. Vincenzi, Padovan-like sequences and generalized Pascal’s Triangles, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, LXVI (f1), 25–35, 2020
  4. H. Belbachir, T. Komatsu and L. Szalay, Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities, Math. Slovaca 64, 287–300, 2014
  5. J.H. Conway and R.K. Guy, The Book of Numbers, World Scientific, Singapore, 2008
  6. A. Fiorenza and G. Vincenzi, Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients, Chaos Solitons Fractals 44, 147–152, 2011
  7. N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Math. Comput. Simulation 125, 168–177, 2016
  8. S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Solitons Fractals 98, 178–182, 2017
  9. A. Ipek, On (p, q)-Fibonacci quaternions and their Binet’ s Formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebras 27 (2), 1343–1351, 2017
  10. G. Kallós, A generalization of Pascal’s triangle using powers of base numbers, Ann. Math. Blaise Pascal 13 (1), 1–15, 2006
  11. T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley–Interscience, New York, 2001
  12. L. Maronhič, Plastic Number: Construction and Applications, 5th Virtual International Conference on Advanced Research in Scientific Areas (ARSA-2016) Slovakia, November 7–11, 2016
  13. L. Németh, The trinomial transform triangle, J. Integer Seq. 21 (7), Art. 18.7.3, pp18, 2018
  14. L. Németh and L. Szalay, Recurrence sequences in the hyperbolic Pascal triangle corresponding to the regular mosaic 4,5, Ann. Math. Inform. 46, 165–173, 2016
  15. M.J. Ostwald, Under siege: the golden mean in architecture, Nexus Netw. J. 2, 75–81, 2000
  16. R. Padovan, Proportion: Science, Philosophy, Architecture,1-st Edition, Taylor and Francis, London and New York, 1999
  17. R. Padovan, Dom Hans van der Laan and the Plastic Number, in:Williams K., Ostwald M. (eds) Architecture and Mathematics from Antiquity to the Future, Birkhäuser, 407–419, 2015
  18. P. Szalapaj, Contemporary Architecture and the Digital Design Process, Routl