Question

1. Call a string of letters "legal if it can be produced by concatenating (running together) copies of the strings 'a', 'b', 'cc' and 'ddd'. For example, the string 'accb' is legal because it can be produced by concatenating 'a', 'cc' and 'b', but the string 'ccca' is not legal. For each integer n 21, let tn be the number of legal strings with n letters. For example, t1 2 (a and 'b' are the legal strings) (a) Write down t2 and a st of all the legal strings of length 2. (b) Write down t3 and a list of all the legal strings of length 3. (c) Find a recurrence for tr that holds for all n 2 4. Explain why your recurrence gives the correct number for tn (d) Use your recurrence to find t4

Answer 1

a)

aa, bb, cc, ab, ba -> t2-5

b)

aaa, bbb, ddd, acc, bcc, cca, ccb, aab, aba, baa, bba,bab, abb -> t3- 13

c) $t_n=2t_{n-1}+t_{n-2}+t_{n-3}$ because we can append either 'a' or 'b' to a string of length (n-1) or append 'cc' to a string of length (n-2) or 'ddd' to a string of length (n-3)' to create a 'legal' string

d) $t_4=2t_{3}+t_{2}+t_{1}=2\cdot 13+5+2={\color{Red} 33}$