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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 17, No. 1, pp. 25-30 (2001)
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On repetitions in frequency blocks of a generalized Fibonacci sequence

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Gyula Darvasi

Department of Mathematics and Computer Science College of Nyíregyháza Po Box 166, Nyíregyháza, Hungary, H-4401 darvasi@zeus.nyf.hu

**Abstract:** The Fibonacci sequence $U_0=1, U_1=5$ and $U_n=3\cdot U_{n-1}+U_{n-2}$ for $n\ge 2$ yields a purely periodic sequence $\{\bar U_n\}=\{U_n\pmod m\}$ with an integer $m\ge 2$. Consider any shortest full period of $\{\bar U_n\}$ and form the number block $B_m\in\mathbb{N}^m$ 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.

**Keywords:** Second-order linear recurrences, uniform distribution.

**Classification (MSC2000):** 11B37; 11B39, 11B50

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