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Question: Let f(n) be the number of bit strings of length n that contain three consecutive 0's. Note: a str...

Let f(n) be the number of bit strings of length n that contain three consecutive 0's. Note: a string that contains four or more consecutive 0's is also considered to contain three consecutive 0's. We want to define f(n) recursively. we consider n >= 5. We want to express f(n) recursively in terms of f(n-1); f(n-2),......

Here are some hints:

(Step 1) If x is a bit string of length n-1 that contains three consecutive 0's, then extending it by a bit 0 or bit 1 will give rise to a string of length n that again contains three consecutive 0's. (Step 2) Consider a bit string x of length n that contains three consecutive 0's, but is not derived by Step 1. Next, answers the following two questions about such string x.

Q1.Compute f(16). verify that the f(16) that you computed is correct by writing a java program to enumerate all 216 = 65536 bit strings of length 16, and count the strings that contain three consecutive 0's.