Duplications in the k-generalized Fibonacci sequences - Luca Florian (NEW YORK JOURNAL OF MATHEMATICS, 2021)
Let k >= 3 be an odd integer. Consider the k-generalized Fibonacci sequence backward. The characteristic polynomial of this sequence has no dominating zero, therefore the application of Baker's method becomes more difficult. In this paper, we investigate the coincidence of the absolute values of two terms. The principal theorem gives a lower bound for the difference of two terms (in absolute value) if the larger subscript of the two terms is large enough. A corollary of this theorem makes possible to bound the coincidences in the sequence. The proof essentially depends on the structure of the zeros of the characteristic polynomial, and on the application of linear forms in the logarithms of algebraic numbers. Then we reduced the theoretical bound in practice for 3 <= k <= 99, and determined all the coincidences in the corresponding sequences. Finally, we explain certain patterns of pairwise occurrences in each sequence depending on k if k exceeds a suitable entry value associated to the pair.