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Maximal frequencies of elements in second-order linear recurring sequences over a finite field. (English) Zbl 0747.11062

Let \(w(a,b)=(w)\) be a second order linear recurring sequence over \(\mathbb{F}_ q\) satisfying the relation \(w_{n+2}=aw_{n+1}-bw_ n\) with initial terms \(w_ 0\), \(w_ 1\). Assuming that \(b\neq 0\), \(w(a,b)\) is purely periodic. If \(d\in\mathbb{F}_ q\), let \(A(d)\) denote the number of times that \(d\) appears in a full period of the sequence \(w(a,b)\). The main result of this paper is to prove that \(A(d)\leq\min(q,2\text{ord}(b))\) for \(d\neq 0\) and \(A(0)\leq\min(q- 1,2\text{ord}(b))\). This result was proved in the special case of \(b=\pm 1\), \(w_ 0=0\), \(w_ 1=1\) by the second and third author before.

MSC:

11T99 Finite fields and commutative rings (number-theoretic aspects)
11B37 Recurrences
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