Textbook Problem

Counting Strings: Consider the set of all string of $a\text{'}s,b\text{'}s,$ and $c\text{'}s.$

1. Make a list of all of these string of lengths 0, 1, 2, and 3 that do not contain the pattern aa.
2. For each integer $n\ge 0,$ let $s_n=$ the number of strings of $a\text{'}s,\text{}b\text{'}s,$ and $c\text{'}s$ of length n that do not contain the pattern aa. Find $s_0,s_1,s_2,$ and $s_3,\mathrm{....}$
3. Find a recurrence relation for $s_0,s_1,s_2,\mathrm{....}$
4. Use the results of parts (b) and (c) to find the number of strings of $a\text{'}s,b\text{'}s,$ and $c\text{'}s$ of length four that do not contain the pattern aa.
5. Use the technique described in Section 5.8 to find an explicit formula for $s_0,s_1,s_2,\mathrm{...}$