The equivalent index j, for j >= 0, of a given index k, for -n <= k < 0, is j = n - k, where n is the length of the sequence.

Suppose we have a sequence of n elements. For the given indices k and j, as defined above, we'll prove by induction that  k + j = n.

For k = -1, j = n - (-1) = n + 1, we have k + j = -1 + n + 1 = n. 
Suppose for  k = i, j = n - i, j + k = i + n - i = n.
Then for k = i + 1, j = n - i - 1, j + k = i + 1 + n - i - 1 = n.

We proved that k + j = n for any sequence of n elements, therefore  j = n - k.
