Let $u_n$ be a sequence of integers satisfying a linear recurrence relation $$u_{n+k}=a_1u_{n+k-1}+a_2u_{n+k-2}+\ldots+a_ku_{n},$$ where the $a_i$ are rational numbers and $k$ is a positive integer. Must the $a_i$ all be integers?
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Counter-example: $$ u_{n+2}=\frac{1}{2}(u_{n+1}+u_n),\quad n\ge0\\ u_0=u_1=2 $$ Every term of this sequence is $2$.
Jack's wasted life
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Thank you for this Jack. I wonder if all counterexamples are so transparently true, or if some such sequences $u_n$ are integer-valued for deep reasons? – falang Oct 29 '16 at 04:49
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@falang: The sequence $u_n$ satisfies in fact the recurrence $u_{n+1} = u_n$. From this recurrence, we can get a lot of recurrences of higher order ( multiply the polynomial $(x-1)$ by other poly). Deeper question still standing. – orangeskid Sep 25 '22 at 21:40