On Lacunary Recurrences

Let {an}^_ao be a sequence which satisfies a linear recurrence of order k +1. We are herein concerned with the lacunary subsequences {®mn+b}Z=-ao> where m and b are fixed integers, so called because they consist of the terms from {aJ with lacunae, or gaps, of length m between them. In [5], [2], and [3] it has been shown that, for any m and A, the subsequences {amn+b} also satisfy a linear recurrence of order k + l. In this note we shall express the coefficients of this recurrence in terms of generalized Dickson polynomials, by means-of their functional equations, and present some applications of this description. As corollaries to our main theorem we give generalizations, to prime power moduli, of the known result ([5], Theorem 4) that whenever/? is prime, the subsequences { a ^ ^ } ^ satisfy the same linear recurrence modulo p as is satisfied by {an}. We conclude with an analog of Howards tribonacci identity ([3], Theorem 3.1) for tetranacci sequences.