The Fibonacci numbers are defined by $\displaystyle F_0=0$,$\displaystyle F_1=1$ and $\displaystyle F_n=F_{n-1}+F_{n-2}$

Prove if $\displaystyle 2|F_n$, then $\displaystyle 3|n$

This is one of those times where it's easy to see intuitively, it's hard to see formally.

Considering the Fibonacci sequence

$\displaystyle 0,1,1,2,3,5,8,13,21,36...$

It falls into a pattern of

$\displaystyle E,O,O,E,O,O,E,O,O,E,...$

From, as the first term $\displaystyle F_0$ is even, it's clear to see that every third term beginning from $\displaystyle F_1$ is divisible by 3, as it is the sum of the 2 previous numbers which are odd.

But this just scaffolding, scrap work.

Any tips on how to codify this formally?