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Stability classes of second-order linear recurrences modulo \(2^k\). II. (English) Zbl 1197.11023
The authors study second-order linear recurring sequences \(w_{i}= aw_{i-1}+ bw_{i-2}\) with integer parameters \(a,b\) and integer initial terms \(w_{0}\) and \(w_{1}.\) The modulo \(m\) reduced sequence is periodic with least period length \(\lambda (m)\). The authors consider the case \(m=p^{k}\) (prime power moduli) and study \(\lambda (p^{k})\) as well as the quantity \(\nu(d, p^{k})\) which denotes the number of times that the residue \(d\) appears in a single period. Let \[ \Omega (p^{k})= \{\nu(d,p^{k}): d \in\mathbb{Z}\} \] and define a sequence \((w_{i})\) to be stable modulo \(p\) if there is a positive integer \(N\) such that \(\Omega (p^{k})= \Omega (p^{N})\) for all \(k \geq N\). The main results of the present paper are concerned with stability properties modulo 2. In the case \(a\) odd and \(b \equiv 5\bmod 8\) a characterization is obtained.
Part I, see Tatra Mt. Math. Publ. 20, 31–57 (2000; Zbl 0992.11013).
MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B50 Sequences (mod \(m\))
11B37 Recurrences
11K36 Well-distributed sequences and other variations
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