On generalized Fibonacci numbers

We provide a formula for the nth term of the k-generalized Fibonaccilike number sequence using the k-generalized Fibonacci number or knacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation x+x−k = 2. We then extend our results to k-generalized Horadam (kGH) and k-generalized Horadam-like (kGHL) numbers. In dealing with the limit of the ratio of successive terms of kGH and kGHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for k-periodic k-nary Fibonacci sequence can also be derived. Mathematics Subject Classification: 11B39, 11B50