On some combinations of terms of a recurrence sequence

Let (Gm)m ≥ 0 be an integer linear recurrence sequence (under some weak technical conditions) and let x   ≥ 1 be an integer. In this paper, we are interested in the problem of finding combinations of the form xGn+Gn−1xGn+Gn−1 which belongs to (Gm)m ≥ 0 for infinitely many positive integers n. In this case, we shall make explicit an upper bound for x which only depends on the roots of the characteristic polynomial of this recurrence. As application, we shall study the k-nacci case.