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On Some Combinations of Non-Consecutive Terms of a Recurrence Sequence
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Eva Trojovská

Department of Mathematics

Faculty of Science

University of Hradec Králové

Czech Republic

**Abstract:**

Let (*G*_{m})_{m≥0} be an integer
linear recurrence sequence (satisfying some weak technical conditions)
and let *x* ≥ 1 be an integer. In this paper, among other things,
we are interested in non-consecutive combinations
*x**G*_{n+a} +
*G*_{n} that belong to the sequence
(*G*_{m})_{m≥0} for infinitely many
positive integers *n*. In this case, we make explicit an upper
bound for *x* that depends only on *a* and the zeros of the
characteristic polynomial of this recurrence (this generalizes previous
papers of Trojovský). As an application, we study the Fibonacci case.

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(Concerned with sequence
A000045.)

Received February 19 2018; revised version received March 11 2018.
Published in *Journal of Integer Sequences*, March 12 2018.

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