About the number of C∞-words of form

Let pi(n) denote the number of the C∞-words of form  with length 2n+i and gap i, where i is the length of word x. In this paper, we prove the following result: pi(n)=6 for all n≥1 and i=2,3,4. Moreover, we provide a complete solution of for |x|≤4.