Solution: 2018 Fall Final - 9

Author: Michiel Smid

Question

Consider bitstrings that do not contain 110. Let $S_n$ be the number of such strings having length $n$. Which of the following is true for any $n \geq 4$?

(a)

$S_n = S_{n-1} + S_{n-2} + S_{n-3}$

(b)

$S_n = S_{n-1} + S_{n-2} + 2^{n-2}$

(c)

$S_n = S_{n-1} + S_{n-2} + 1$

(d)

$S_n = S_{n-1} + S_{n-2} + 2^{n-3}$

Solution

Let’s write out the possibilities and sum them

$ 0, S_{n-1} $

$ 1, 0, S_{n-2} $

We also have to consider the possibility of having all 1’s, which isn’t considered in the above possibilities

$ S_n = S_{n-1} + S_{n-2} + 1 $