Arithmetic index subsequences of linear recurrences

In this paper, we study the subsequences of homogeneous linear recurrences of order [Formula: see text] such that a subsequence consists of every [Formula: see text]th ([Formula: see text]) term of the original sequence. There exist general and sporadic results on this area, the main advantage of our method is that we use only the coefficients of the input sequence to find the coefficients of the claimed subsequence explicitly. We do not use any intermediate object to formalize the final results. The two approaches we suggest work, at least in theory, for any given [Formula: see text] and [Formula: see text], but the computation becomes more and more technical and difficult for larger values. In this work, we handle explicitly the cases [Formula: see text] for arbitrary [Formula: see text], and [Formula: see text] with arbitrary [Formula: see text]. Moreover, we give some information to deal with the problem of [Formula: see text]. The computational results are displayed in clear tables. The main tool we apply is the theory of symmetric polynomials.