Question

Find a recurrence relation for an, the number of sequences of length n formed by u's, v's, and w's with the subsequence vv not allowed.

b) (5 pts) Repeat part a) but now with the requirement that there is no subsequence uwv.

edit: thats all i have for this question

Answer 1

(a)

According to Question:-

No two subsequence vv not allowed so,

A) ends with v or w or

B) ends with u

The Type (A) good sequences of length n are obtained by appending v or w to a good sequence of length n−1. So there are 2a(n−1) Type (A) good sequences of length n.

The Type (B) good sequences of length n ≥ 2 are obtained by appending u to a good sequence of length n−1 that doesn't end in u. Such a good sequence is obtained by appending v or w to a good sequence of length n−2, so there are 2a(n−2) of them.

It follows that for n≥2,

a(n)= 2a(n−1) + 2a(n−2)

As initial conditions we can use a(0) = 1 ,  a(1) = 3.

since the possibilities are {u},{v},{w}, and a(2) = 8. Then I can solve for a(3) by accounting for the total number of combinations, 3 pow(3),

then subtracting cases of adjacent u's, totaling to 9 cases,

which gives the result of 18.

(b)

According to Question:-

since the possibilities are {u},{v},{w}, and a(2) = 8. Then I can solve for a(3) by accounting for the total number of combinations, 3 pow(3),

Now , ther is only 3 cases ,

which gives the result of 24.