Integers represented by binary recursive sequences

TL;DR

This paper investigates integers represented by binary recursive sequences, providing bounds on the index and growth of sequence terms based on initial values, with some bounds proven to be optimal apart from constants.

Contribution

It introduces new bounds on the highest index with zero terms and on the growth rate of sequence terms, depending solely on initial values, for non-degenerate binary recursive sequences.

Findings

Bounds for the highest index with zero terms

Bounds on the growth order of sequence terms

Some bounds are proven to be optimal apart from constants

Abstract

This paper is the continuation of \cite{htl}, where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In case of non-degenerate sequences we give bounds for the highest index for which a term can be 0 and bounds on the growth order of the absolute values of the terms, both only in terms of the two initial values, which is a novel feature. Some of these bounds are best possible apart from a multiplicative constant.