On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers

Let $(F_{r}^{(k)})_{r\geq2-k}$ and $(L_{r}^{(k)})_{r\geq2-k}$ be generalizations of the Fibonacci and Lucas sequences, where $k\geq2$. For these sequences the initial $k$ terms are $0, 0, \ldots , 0, 1$ and $0, 0, \ldots , 2, 1$, and each subsequent term is the sum of the preceding $k$ terms. In this paper, we determined all first and second kinds of Thabit numbers that can be expressed as the sums of $k$-Fibonacci and $k$-Lucas numbers. We employed the theory of linear forms in logarithms of algebraic numbers and a reduction method based on the continued fraction.