The K-periodic Fibonacci Sequence and an Extended Binet’s Formula

It is well-known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational α. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r-Fibonacci sequences, and the Pell sequence, to name a few. We show that these sequences satisfy a linear recurrence relation when considered modulo k, even though the sequences themselves do not. We then employ this recurrence relation to determine the generating functions and Binet-like formulas. We end by discussing the convergence of the ratios of the terms of these sequences.