Skinny Multiplication Table

This is a puzzle.
Imagine a look-up table like this:

\begin{matrix} k & \quad &             0  & 1  & 2  & 3  & 4  & 5 \dots \\\\ magic[k]      & \quad & 0  & 0  & 1  & 2  & 4  & 6 \dots \end{matrix}

and that this table is used for multiplication. Obviously, this multiplication table must be used differently from the usual one, because it is a one-dimensional table, long and skinny.

To find 6 \times 3 , the table would be consulted for both 6 + 3 and 6 - 3 , and the results subtracted from each other.

6 \times 3 = magic[6+3] - magic[6-3]. \quad \quad \quad \quad (*)

To find 3 \times 6 , we could either simply swap the numbers before using the table, or extend the table towards the left.

I’ve included enough magic numbers in the table so you can check that the scheme works for 3 \times 0 , 4 \times 1 , 2 \times 2 and 3 \times 2 .

Can you reconstruct the values in the table, e.g. to find what magic[21] must contain? Can you demonstrate that the scheme works or prove that it doesn’t work? Can you construct a magic table different from mine that also allows (*) to work?

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