At https://www.maa.org/math-competitions/putnam-competition it says, "The Putnam Competition covers a range of material in undergraduate mathematics, including elementary concepts from group theory, set theory, graph theory, lattice theory, and number theory," so all you have to do is trundle off to the library/bookstore to find books that cover those topics....


Here's a simple though non-geometric demonstration of period $6$: $x_{n+6}=x_{n+5}-x_{n+4}=x_{n+4}-x_{n+3}-x_{n+4}$ $=-x_{n+3}=-x_{n+2}+x_{n+1}=-x_{n+1}+x_{n}+x_{n+1}=x_n$

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