Question

a) Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven do not contain three consecutive 0s?

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a) Let ana_{n} represent the number of bit strings of length n that don't contain a 3 consecutive 0's.

an=an1+an2+an3a_{n} = a_{n-1} + a_{n-2} + a_{n-3} for n3n \geq 3.

If the string starts with 1, then there are n1n-1 different bit strings without 3 consecutive 0's. Therefore, an1a_{n-1}. Continue with bit strings starting with 01 and 001.

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