On some new results for the generalised Lucas sequences

Abstract

In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

Keywords

• generalised lucas sequence
• generalised pell-lucas sequence
• pell equation
• special pell equation
• negative pell equation
• primary 11b83
• secondary 26c05