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On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Author

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  • Eva Trojovská

    (Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic)

  • Pavel Trojovský

    (Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic)

Abstract

Let ( t n ( r ) ) n ≥ 0 be the sequence of the generalized Fibonacci number of order r , which is defined by the recurrence t n ( r ) = t n − 1 ( r ) + ⋯ + t n − r ( r ) for n ≥ r , with initial values t 0 ( r ) = 0 and t i ( r ) = 1 , for all 1 ≤ i ≤ r . In 2002, Grossman and Luca searched for terms of the sequence ( t n ( 2 ) ) n , which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ ≥ 1 , there exists an effectively computable constant C = C ( ℓ ) > 0 (only depending on ℓ ), such that, if ( m , n , r ) is a solution of t m ( r ) = n ! + ( n + 1 ) ! + ⋯ + ( n + ℓ ) ! , with r even, then max { m , n , r } < C . As an application, we solve the previous equation for all 1 ≤ ℓ ≤ 5 .

Suggested Citation

  • Eva Trojovská & Pavel Trojovský, 2021. "On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers," Mathematics, MDPI, vol. 9(9), pages 1-9, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:962-:d:543292
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    References listed on IDEAS

    as
    1. Pavel Trojovský, 2019. "On Terms of Generalized Fibonacci Sequences which are Powers of their Indexes," Mathematics, MDPI, vol. 7(8), pages 1-10, August.
    2. 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.
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