ON REPETITIONS IN FREQUENCY BLOCKS OF A GENERALIZED FIBONACCI SEQUENCE

Abstract. The Fibonacci sequence U0 = 1, U1 = 5 and Un = 3 ·Un−1+Un−2 for n ≥ 2 yields a purely periodic sequence {Ūn} = {Un (mod m)} with an integer m ≥ 2. Consider any shortest full period of {Ūn} and form the number block Bm ∈ Nm to consistof the frequency values of the residue d when d runs through the complete residue systyem modulo m The purpose of this paper is to show that such frequency blocks can nearly always be produced by repetition of some multiple of their first few elements a certain number of times. Theorems 1 and 2 contain our main results where we show when this repetition does occur, what elements will be repeated, what is the repetition number and how to calculate the value of the multiple factor. Let U0 = 1, U1 = 5 and define the recurrence {Un} = U(3, 1) to satisfy the relation Un = 3 · Un−1 + 1 · Un−2 for n ≥ 2. For an integer m ≥ 2, let {Ūn} denote the sequence {Un} considered modulo m. It is known that {Ūn} is purely periodic [7], that is, there exists a positive integer r such that Ūn+r = Ūn for all n ∈ N. Define h(m) to be the length of a shortest period of {Ūn}, and S(m) t