A ternary string is a string character from the set 012 Recu

A ternary string is a string character from the set {0,1,2}. Recursivley define the set of ternary strings with no consecutive 0\'s.

Solution

The a recurrence relation for the number of ternary strings of length n that contain a pair of consecutive 0\'s is:

      a_n =2a_n?1 +2a_n-2a n?2 +3^n?2

      and

The a recurrence relation for the number of ternary strings of length n that contain a pair of no consecutive 0\'s is:

     i,e The number b_n   of ternary strings of length n that do not contain the substring 00 is actually

                          b_n=3ⁿ-a_n

                                            =3ⁿ-(2a_n?1 +2a_n-2a n?2 +3^n?2)

                                =3ⁿ ?(2(3^n?1 ?b_n?1 )+2(3^n?2 ?b_n?2 )+3^n?2 )

                                 =3ⁿ ?(2.3^n?1+2.3^n?2  ?2.b_n?1 ?2.b_n?2

                                               =2b_n-1+2b_n-2.

                                 =3 n ?(2?3 n?1 +3?3 n?2 ?2b n?1 ?2b n?2 )=2b n?1 +2b n?2 .