On the Distribution of a Certain Family of Fibonacci Type Sequences

Taking the Fibonacci sequence G0 = 1; G1 = b 2 f1; 3; 5g and Gn+1 = 3 Gn + Gn 1 (n 1) with an integer 2 m 2 N , we get a purely periodic sequence fGn( mod m)g. Consider any shortest full period and form a frequency block Bm 2 N to consist of the frequency values of the residue d when d runs through the complete residue system 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 rst few elements a certain number of times. Theorems 3,4 and 5 contains 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. Let A;B 6= 0; G0 = a;G1 = b with a + b > 0 be xed rational integers, let D = A 4B 6= 0 and de ne the Fibonacci type sequence fGng = G(A;B; a; b) to satisfy the recurrence relation Gn+1 = A Gn B Gn 1 for n 1: Let fUng and fVng be the Fibonacci and the Lucas sequence deriving from fGng for a = 0; b = 1 and for a = 2; b = A, respectively. Then it is easy to check Gn = a 2Vn+(b a2A) Un. For r1;2 = A p D 2 the following equations hold (1) Un = (r n 1 r 2 )=(r1 r2) and Vn = r 1 + r 2 ;