If you like this kind of stuff, you might try to understand the following pattern (F_n is Fibonacci, F_0 = 0, F_1 = 1).
F_n%2==0 whenever n%3==0.
F_n%3==0 whenever n%4==0.
F_n%4==0 whenever n%6==0.
F_n%5==0 whenever n%5==0, etc.
Ex. F_n mod 2:
1, 1, 0 (as 1+ 1 = 2 == 0 in mod 2), 1, 1, 0, ...
So on and so forth.
If you like this kind of stuff, you might try to understand the following pattern (F_n is Fibonacci, F_0 = 0, F_1 = 1).
F_n%2==0 whenever n%3==0.
F_n%3==0 whenever n%4==0.
F_n%4==0 whenever n%6==0.
F_n%5==0 whenever n%5==0, etc.