On some combinations of terms of a recurrence sequence.

Let ( G m ) 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 x G n + G n − 1 which belongs to ( G m ) 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. [ABSTRACT FROM AUTHOR]