Recurrence relation for number of ternary strings that contain two consecutive 2s

Question

Q: A string that contains only 0s, 1s and 2s is called a ternary string.

Find a recurrence relation for the number of ternary strings of length n that contain two consecutive 2s.

I don't really understand the question, can someone help me?

Answer 1

The question means: if $T_2(n)$ is the number of strings of length $n,$ which contain two consecutive twos then find some function $f$ such that $T_2(n) = f(T_2(n-1), \dotsc, T_2(1)).$

Answer 2

You define $A(n)$ as the number of ternary strings of length $n$ that contain two $2$s in a row. If you didn't have the restriction there would be $3^n$ ternary strings of length $n$. However, $A(0)=A(1)=0$ because there isn't room for two $2$s in a row. $A(2)=1$ because the only length $2$ string that contains two $2$s in a row is $22$. You are expected to find an equation of the form $A(n)=aA(n-1)+bA(n-2)+$ maybe some more terms by thinking about how you can form a string of length $n$ that has two $2$s. One way is to start with a string of length $n-1$ that already has two $2$s in a row and append any digit.