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One of my favourite examples in this context is the following: Define a sequence $(s_n)$ by $s_1=8$, $s_2=55$ and for $n\geq3$ $s_n$ the smallest integer such that $s_n/s_{n-1}>s_{n-1}/s_{n-2}$ so that $s_3=379$ as $379/55>55/8$. Then we have $s_n=6s_{n-1}+7s_{n-2}-5s_{n-3}-6s_{n-4}$ for $5\leq n\leq11056$ but not for $n=11057$ (I have lost track of the name of the person to whom this is due, but it is, nowadays, easily verified on a computer).