Does this sequence always terminate or enter a cycle? I've been fiddling with the recursive sequence defined as follows:
$$\begin{equation}
  f_n=\begin{cases}
    a, & n=1.\\
    b, & n=2.\\
c, & n=3.\\
f_{n-1}f_{n-2}f_{n-3} \mod[f_{n-1}+f_{n-2}+f_{n-3}], & n>3.
  \end{cases}
\end{equation}$$
And no matter my initial choices of positive integers $a,b,c$, it seems $ \{  f_n \}$ always terminates (three consecutive zeros) or enters a cycle. For instance, if $a=12,b=12,c=9$, then the sequence becomes $12,12,9, 9,12,$ $12\dots$
My question: can we prove (or disprove) that for any positive integers $a,b,c$, the sequence $\{ f_n\}$ will always terminate (three consecutive zeros) or enter a cycle? 

More important remarks in quotes.
(Dec. 27) Remark 1.1: it appears (though I have not proved it) that my conjecture is true for the simpler recursive sequence 
$$\begin{equation}
  f_n=\begin{cases}
    a, & n=1.\\
    b, & n=2.\\
f_{n-1}f_{n-2} \mod[f_{n-1}+f_{n-2}], & n>2.
  \end{cases}
\end{equation}$$
Perhaps this would be a better starting point. Hereafter, I will only be referring to the above.
(Dec. 28) Remark 1.2: If $f_n=f_{n-1}$ and is odd, then $f_k=f_n$ for all $k>n$. If $f_n=f_{n-1}$ and is even, the sequence will terminate. This can be proven simply enough.

(Dec. 28) Remark 1.3: I conjecture that if $a$ is odd and $b>a+1$ is even, the sequence always terminates. Also, if $a$ is even and $b>a+1$ is odd, the sequence never terminates.

(Dec. 28) Remark 1.4: the sequence reaches the cycle $\dots 5,7,11,5,7,11\dots$ for many choices of $a,b.$ Some pairs $(a,b)$ for which $f_n$ enters the cycle $\dots 5,7,11 \dots$ are $(3,5), (5,7), (7,11),$ $(7,3),(35,11),(44,13).$ There are probably infinitely many pairs $(a,b)$ for which this occurs. The frequency with which I see $5,7,11$ is probably due to my relatively small choices of integers. I wonder what the minimum of $X+Y+Z > 3$ is, where $X,Y,Z$ is a cycle eventually reached by the function.
I wonder further if there are arbitrarily long sequences of numbers which this recurrence relation would cycle through for certain initial $a,b$. I have not found any cycles longer than three terms, though $5,7,11$ is not the only three-term cycle I have found. For $(a,b) = (7,111111101)$, the sequence eventually reaches the cycle $8496495, 3641355, 6068925$. If we have $(a,b) = (6, 99)$, the sequence reaches a different length-$3$ cycle.
(Dec. 28) Remark 1.5: almost always, it seems that when $f_n$ does not terminate, the repeated terms are multiples of $5$. Some exceptions are $\{ f_n \}^{(9,66)}$, $\{ f_n \}^{(6,99)}$, and $\{ f_n \}^{(3,11)}$, where $\{ f_n \}^{(x,y)}$ is the sequence generated for $a=x,b=y$.

(Dec. 28) Remark 1.6: I conjecture $5,7,11$ are the only primes that appear in a distinct cycle (see Def. 1.1). In fact, it may even be the case that $5,7,11$ is the only distinct cycle with primes.

(Dec. 29) Remark 1.7: I should probably state what the 'cycles' I am talking about are.
Definition 1.1: $\{ f_n \}$ is said to enter cycle $X,Y,Z$ if for some $k>0$ we have $f_{k+3n} = X, f_{k+3n+1} =Y$, and $f_{k+3n+2} = Z$ for all integer $n \geq 0$.
Definition 1.2: A cycle is said to be non-constant if $X,Y,Z$ are not all equal. Similarly, a cycle is said to be distinct if $X \neq Y \neq Z$.

(Dec. 29) Remark 1.8: It seems not every positive integer is part of a distinct cycle (see Def. 1.2) That is, there are some (in fact, many) integers for which, no matter our choice of integers $a,b > 0$, the sequence $\{ f_n \}^{(a,b)}$ will not enter a distinct cycle with that integer. I am not sure if the same is true for non-constant cycles. For constant cycles, this is trivially not the case.


I think there can be cycles of any odd length.
Take $2n+1$ odd primes $p_k$, for which any $n$ of them have a sum less than the other $n+1$.
Let $a_i = p_i-p_{i+1}+p_{i+2}-....+p_{i-1}$, where the index is taken cyclically.  Then $a_i+a_{i+1}=2p_i$, and all the $a_i$ are odd.
Let $N$ be an odd number for which $Na_ia_{i+1}=a_{i+2}\pmod{a_i+a_{i+1}}$.  That is possible by the Chinese Remainder Theorem.
Then take the numbers $\{Na_i\}$ as the cycle.
