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

Abstract

ABSTRACT Let k ≥ 2 be a fixed integer. The k-generalized Lucas sequence { L n ( k ) } n ≥ 0 \[{{\left\{ L_{n}^{\left( k \right)} \right\}}_{n\ge 0}}\] starts with the positive integer initial values k, 1, 3, …, 2 k−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 \[\left| n-m \right|<\sqrt{m}\] . In this paper, we combine these two concepts. We solve completely the diophantine inequality | L n ( k ) − 2 m | < 2 m / 2 \[\left| L_{n}^{\left( k \right)}-{{2}^{m}} \right|<{{2}^{m/2}}\] 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 \[L_{n}^{\left( k \right)}={{2}^{m}}+t\] with the condition |t| < 2 m/2, t ∈ ℤ \[t\in \mathbb{Z}\] . We also discovered a new formula for L n ( k ) \[L_{n}^{\left( k \right)}\] which was very useful in the investigation of one particular case of the problem.