TITLE: Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?
QUESTION [17 upvotes]: About this Fibonacci sequence, is it true that the remainders when divided by three repeat along with the sequence like this:
Fibonacci sequence:
$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946\cdots$
Remainder when divided by $3$ sequence:
$1, 1, 2, 0, 2, 2, 01, 00, 01, 01, 02, 000, 002, 002, 001, 000, 0001, 0001, 0002, 0000, 00002\cdots$
I made each one the same number of digits to align them to avoid confusion.  Is it true that it happens every eight numbers?  If so, how and why does this happen?

REPLY [2 votes]: Consider the pairs $(R_k,R_{k+1})$ with $R_k$ is the remainder of $F_k$ when dividing by $3$. Since there are infinitely many such pairs, by Pingeonhole principle there exists $k,l$ such that $(R_k,R_{k+1})=(R_l,R_{l+1})$. Thus $R_{k-1}=R_{l-1}$ and so on. So the remainders are periodic.
The same property is true for all sequence with linear recurrence (with the same method).