Alternate Terms in Lucas Sequence

Abstract

In this paper we study the properties of L_2n+1 which is the Lucas number of order 2n+1. Several properties like generating functions, recurrence relations, summation formulas and (q-analogues of L_2n+1 were found by Agarwal in [1,2,3]. Here we obtain hypergeometric form, Integral representation and several congruence properties and identities for these numbers. Congruence properties are used to establish a theorem on periodicity of the sequence {L_2n+1}_n^∞=0.