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 = F q [ 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 g 0 = 0, g 1 = 1 and gn + 1 = gn + D gn - 1 for n > 1, where D is any nonzero polynomial in R = F q [ t ] A new identity is established.