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

$a_{n} = a_{n-1} + a_{n-2} + a_{n-3}$ for $n \geq 3$.

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

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