TUYMAADA 2006 Problem 2

We call a sequence of integers a Fibonacci-type sequence if it is infinite in both ways and $a_{n}=a_{n-1}+a_{n-2}$ for any $n\in\mathbb{Z}$ . How many Fibonacci-type sequences can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than $N$ ? (two sequences are considered to be the same if they differ only by shifting of indices)

Proposed by I. Pevzner