Keywords:-

Keywords: Fibonacci Sequence, Generalized Recurrence, Diophantine Equations, Exponential Bounds, Number Theory, Linear Recurrences

Article Content:-

Abstract

Generalized Fibonacci sequences, also known as k-bonacci sequences, are extensions of the classical Fibonacci sequence that incorporate k previous terms into their recurrence relation. These sequences have wide applications in number theory, combinatorics, computer science, and applied mathematics. The present paper provides an explicit exponential bound for the growth of k-generalized Fibonacci numbers, expressed in terms of the dominant root αₖ of the corresponding characteristic polynomial. The derived inequality simplifies asymptotic estimates and offers practical benefits for Diophantine analysis and recurrence-based modeling. Through analytical reasoning and numerical evaluation, the study validates the bound for various k values, demonstrating its precision and general applicability. Furthermore, an application to the finiteness of integer representations as linear combinations of k-bonacci numbers with bounded coefficients is presented. This research emphasizes an accessible, rigorous, and computationally verifiable approach to understanding generalized recurrence growth while providing new avenues for mathematical exploration and practical implementation.

References:-

References

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Gargote, A., Gore, H., & Khairkhar, V. (2026). A New Bound on The Growth of K-Generalized Fibonacci Sequences and An Application to Linear Diophantine Representations. International Journal Of Mathematics And Computer Research, 14(03), 88-90. https://doi.org/10.47191/ijmcr/v14iSPC3.18