Is there a non-trivial property that holds for the first 10,000 positive integers, then fails?

Consider the sequences defined by:
s(1) = a; s(2) = b; s(n) = least integer such that s(n)/s(n-1) > s(n-1)/s(n-2).
In other words, s(n) = 1+floor(s(n-1)^2/s(n-2)) for n >= 3.  These
sequences are similar in some ways to the classically-studied Pisot
sequences.  For example, if a = 1, b = 2, then we get the odd-indexed
Fibonacci numbers.

D. Boyd of UBC, an expert in Pisot sequences, pointed out the following.
If we let a = 8, b = 55 in the definition above, then the resulting
sequence s(n) appears to satisfy the following linear recurrence
of order 4:

        s(n) = 6s(n-1) + 7s(n-2) - 5s(n-3) - 6s(n-4)

Indeed, it does satisfy this linear recurrence for the
first 11,056 terms.  However, it fails at the 11,057th term!
And s(11057) is a 9270 digit number.
(The reason for this coincidence depends on a remarkable fact
about the absolute values of the roots of the polynomial
x^4 - 6x^3 - 7x^2 + 5x + 6.)