I came across this sequence in a programming contest F(n)= F(n-1)-F(n-2); Given F0 and F1 find nth term
(http://codeforces.com/contest/450/problem/B) (the contest is over)
Now the solution of this problem is like this The sequence take value f0, f1, f1-f0, -f0, -f1, f0 - f1 then again f0 and the whole sequence is repeated.
I get that this value is being repeated but could not found the reason for this cyclic order. I searched for cyclic order and sequences but could not find sufficient material that could give the actual feel for the reason of the cycle.
Best How To :
Another way to approach this problem. Note that F(n) = F(n - 1) - F(n - 2) is the same as F(n) - F(n - 1) + F(n - 2) = 0 which makes it a linear difference equation. Such equations have fundamental solutions a^n where a is a root of a polynomial: suppose F(n) = a^n, then a^n - a^(n - 1) + a^(n - 2) = (a^2 - a + 1)*a^(n - 2) = 0, so a^2 - a + 1 = 0 which has two complex roots (you can find them) which have modulus 1 and argument pi/3. So their powers 1, a, a^2, a^3, ... travel around the unit circle and come back to 1 after 2 pi/(pi/3) = 6 steps.
This analysis has the same defect as the corresponding one for differential equations -- how do you know to look for solutions of a certain kind? I don't have an answer for that, maybe someone else does. In the meantime, whenever you see a linear difference equation, think about solutions of the form a^n.