Distribution Property of Recursive Sequences Defined

We shall consider a distribution property of sequences of integers. Let us denote a (a„)nGM an infinite sequence of integers. For integers N > 1, m > 2, and j (0 < J < 77Z — 1), let us define AN(j9 m9 a) as the number of terms among al3 a2S ...9aN satisfying the congruence an = j (mod m) . A sequence a (an)nEH±s said to be uniformly distributed modulo m (u.d. mod m) if, for every J = 0, 1, ...,772-1, AN(j9 m9 a) 1 lim = -. (1.1) iu^co N m A sequence a = ( a n ) n € N is said to be uniformly distributed in Z if, for any integer m > 2, a = ( a n ) n e ^ is uniformly distributed modulo 777. This notion was first introduced by Niven [6] and various results are already obtained (see Kuipers & Niederreiters book [4]), among which the sequence of Fibonacci numbers and its generalizations were investigated with respect to uniform distribution property modulo 777. The sequence of generalized Fibonacci numbers is defined by the following linear recurrence formula of second order,