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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
##### Keywords:
Lucas; Fibonacci; distribution; stability
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##### References:
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