The k-Generalized Lucas Numbers Close to a Power of 2.

Let k ≥ 2 be a fixed integer. The k-generalized Lucas sequence { L n (k) } n ≥ 0 starts with the positive integer initial values k, 1, 3, ..., 2k−1 – 1, and each term afterward is the sum of the k consecutive preceding elements. An integer n is said to be close to a positive integer m if n satisfies | n − m | < m . In this paper, we combine these two concepts. We solve completely the diophantine inequality | L n (k) − 2 m | < 2 m / 2 in the non-negative integers k, n, and m. This problem is equivalent to the resolution of the equation L n (k) = 2 m + t with the condition |t| < 2m/2, t ∈ ℤ . We also discovered a new formula for L n (k) which was very useful in the investigation of one particular case of the problem. [ABSTRACT FROM AUTHOR]