The solution involves trying every possible prefix of the sequence and checking its cost. To make a sequence with the minimal cost, you can add $-1,-2,\dots ,-2,-1$ to the sequence to reduce it to the problem of making all the elements in a sequence the same. This is a classical problem where you just change all the elements to the median (proof is left as an exercise for the reader). This algorithm is $\mathcal{O}(N^2)$, but you can use data structures such as a segment tree to speed it up to $\mathcal{O}(N{\log }^{2}N)$.

Time complexity: $\mathcal{O}(N{\log }^{2}N)$