Given that the function f (x) defined on the positive integer set satisfies the condition: F (1) = 2F (2) = - 2F (n + 2) = f (n + 1) - f (n), then the value of F (2011) is?

Given that the function f (x) defined on the positive integer set satisfies the condition: F (1) = 2F (2) = - 2F (n + 2) = f (n + 1) - f (n), then the value of F (2011) is?

∵ f (1) = 2, f (2) = - 2, f (n + 2) = f (n + 1) - f (n), ∵ f (3) = f (2) - f (1) = - 4f (4) = f (3) - f (2) = - 2F (5) = f (4) - f (3) = 2F (6) = f (5) - f (4) = 4f (7) = f (6) - f (5) = 2F (8) = f (7) - f (6) = - 2. F (n) circulates by 2, - 2, - 4, - 2,2,4, and the cycle period is 6 ∵ f (2011) = f (335 × 6 + 1) = f (1

Given K ∈ n +, let F: n + → n + satisfy: for any positive integer n: F (n) = n-k greater than k, please answer and give the reason:
(1) Let k = 1, then the value of one of the functions f at n = 1 is______ .
(2) Let k = 4, and when n ≤ 4, 2 ≤ f (n) ≤ 3, then the number of different functions f is________ .
Correction: (1) let k = 1, then the function value of one of the functions f at n = 1 is.

The problem implies that for a positive integer n less than or equal to K, its function value should also be a positive integer, but the corresponding rule depends on the meaning of the problem. (1) n = k = 1, the condition "a positive integer n greater than k" given in the problem is not suitable, but the function value must be a positive integer, so the value of F (1) is a constant (positive integer); (2) k =