We can use dynamic programming to solve this task. Let $A_i$ be the color of the $i^{\text{th}}$ marble. Let our state be described with 3 items: $(l,r,count)$ that gives the minimal number of marbles we need to insert into subsequence $\{A_l,A_{l+1},\dots ,A_r\}$ so that it disappears, with the added bonus that we can insert $count$ copies of marble $A_l$ on the beginning of the sequence.

For example, let $A=\{\underset{―}{1,3,3,3,2},3,1,1,1,3\}$. State $[\underset{―}{0}][\underset{―}{4}][\mathbf{3}]$ denotes the minimal number of marbles we need to insert into $\{\mathbf{1},\mathbf{1},\mathbf{1},1,3,3,3,2\}$ so that the entire subsequence disappears.

There are three transitions for each state:

• Add a duplicate of the first marble to the beginning of the sequence. This gives $[l][r][X]=[l][r][X+1]+1$.
• If $X=k-1$, we can delete all marbles on the beginning of the sequence, including the first marble of our subsequence. This gives $[l][r][X]=[l+1][r][0]$ for $X=k-1$.
• Third and most complex case is merging the first marble with some later marble. We expect to merge our first marble ($A_l$) with some marble $A_j$ assuming $A_l=A_j$. This merger gives $[l][r][X]=[l+1][j-1][0]+[j][r][X+1]$.

For each state, we must of course find the minimum of all possible transitions.