Abstract
For an integer k ≥ 2, let \((L_{n}^{(k)})_{n}\) be the k-Lucas sequence which starts with 0,…,0,2,1 (k terms) and each term afterwards is the sum of the k preceding terms. In 2000, Luca (Port. Math. 57(2) (2000) 243–254) proved that 11 is the largest number with only one distinct digit (the so-called repdigit) in the sequence \((L_{n}^{(2)})_{n}\). In this paper, we address a similar problem in the family of k-Lucas sequences. We also show that the k-Lucas sequences have similar properties to those of k-Fibonacci sequences and occur in formulae simultaneously with the latter.
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Acknowledgements
One of the authors (JJB) was partially supported by CONACyT from Mexico and Universidad del Cauca, Colciencias from Colombia, and the author (FL) was supported in part by a Marcos Moshinsky Fellowship, Project PAPIIT IN104512, UNAM, Mexico and Project MEC 80120032, CONICYT, Chile. FL also thanks the Department of Mathematics of the University of Valparaiso, Chile for their hospitality during the period when this paper was being written.
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BRAVO, J.J., LUCA, F. Repdigits in k-Lucas sequences. Proc Math Sci 124, 141–154 (2014). https://doi.org/10.1007/s12044-014-0174-7
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DOI: https://doi.org/10.1007/s12044-014-0174-7
Keywords
- Generalized Fibonacci and Lucas numbers
- lower bounds for nonzero linear forms in logarithms of algebraic numbers
- repdigits