Sequences that are eventually periodic for all mod k Suppose we have a sequence of monotone increasing integers $$A=z_1,z_2,z_3...$$ such that for all $k\in\mathbb{N}$, $B_n=mod(A_n,k)$ is eventually periodic. Can we say that $A$ is produced by some recurrance relation $$F_n=\sum_{i=1}^{m}a_iF_{n-i}^{p_i}$$ where the initial points are integers and $p_i\geq 0$.
For example, the Fibonnaci Numbers satisfy the above recurrance relation because they produce a sequenc of integers that are eventually periodic for all $k\in\mathbb{N}$ and $$F_n=\sum_{i=1}^{m}a_iF_{n-i}^{p_i}=\sum_{i=1}^{2}a_iF_{n-i}^{p_i}=1*F_{n-1}^1+1*F_{n-2}^1=F_{n-1}+F_{n-2}.$$ In fact, the length of the period goes by the name 'Pisano Period'.
What I've done: Basically, I haven't made any headway in showing that a sequence of integers with this property must formed by this recurrance relation. The only "progress" has been in showing that the only other sequence of this form that I could think of (integers to the nth power) is in fact $$F_n=\sum_{i=1}^{m}a_iF_{n-i}^{p_i}=\sum_{i=1}^{1}a_iF_{n-i}^{p_i}=aF_{n-1}^1=a^{n}$$ for $F_1=a$ (this can be easily shown).
Overall, as this is for some of my personal research, I would greatly appreciate if anyone who has seen anything about this to post a link or provide a description. The hope is that there is some term or idea that I have not explored yet that can shine some light on this problem.
 A: This is a very natural question, and unfortunately the answer is no. A simple counterexample is $F_n = n!$, which is not only eventually periodic but even eventually constant $\bmod m$ for any modulus $m$. It has the wrong growth rate to be produced by any recurrence of the form you describe; faster than exponential (which is what you get if all of the $p_i$ are at most $1$) but slower than exponential-exponential (which is what you get if at least one of the $p_i$ is at least $2$, and some other mild conditions to rule out the sequence just being periodic). 
A: I found experimental evidence that:


*

*The sequence of restricted Stirling numbers of the second kind A000085 modulo m is eventually periodic for any positive integer m. The lengths of the periods are given by http://oeis.org/A000265
A000085     Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells. 


*The sequence of Fubini numbers A000670 modulo m is eventually periodic for any positive integer m. The lengths of the periods are given by https://oeis.org/A002322 except at powers of 2.


A000670     Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n]. 
