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

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  • 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
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    References listed on IDEAS

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    1. Belbachir, Hacène & Németh, László & Szalay, László, 2016. "Hyperbolic Pascal triangles," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 453-464.
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    3. Alberto Fiorenza & Giovanni Vincenzi, 2013. "From Fibonacci Sequence to the Golden Ratio," Journal of Mathematics, Hindawi, vol. 2013, pages 1-3, March.
    4. Halici, Serpil & Karataş, Adnan, 2017. "On a generalization for fibonacci quaternions," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 178-182.
    5. 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.
    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.
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