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?
Remark: it seems 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.
from Hot Weekly Questions - Mathematics Stack Exchange
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