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On explicit formulae and linear recurrent sequences

Author(s): R. Euler | L. H. Gallardo

Journal: Acta Mathematica Universitatis Comenianae
ISSN 0862-9544

Volume: LXXX;
Issue: 2;
Start page: 213;
Date: 2011;
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Keywords: Polynomials | euclidean division | finite fields | even characteristic.

We notice that some recent explicit results about linear recurrent sequences over a ring R with 1 were already obtained by Agou in a 1971 paper by considering the euclidean division of polynomials over R. In this paper we study an application of these results to the case when R = Fq[t] and q is even, completing Agou's work. Moreover, for even q we prove that there is an infinity of indices i such that gi = 0 for the linear recurrent, Fibonacci-like, sequence defined by g0 = 0, g1 = 1 and gn + 1 = gn + D gn - 1 for n > 1, where D is any nonzero polynomial in R = Fq[t] A new identity is established.
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