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}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Bmatrix%7D+k+%26+%5Cquad+%26+++++++++++++0++%26+1++%26+2++%26+3++%26+4++%26+5+%5Cdots+%5C%5C%5C%5C+magic%5Bk%5D++++++%26+%5Cquad+%26+0++%26+0++%26+1++%26+2++%26+4++%26+6+%5Cdots+%5Cend%7Bmatrix%7D+&bg=ffffff&fg=333333&s=0 "\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 = magic[6+3] - magic[6-3]. \quad \quad \quad \quad (*)](http://s0.wp.com/latex.php?latex=6+%5Ctimes+3+%3D+magic%5B6%2B3%5D+-+magic%5B6-3%5D.+%5Cquad+%5Cquad+%5Cquad+%5Cquad+%28%2A%29+&bg=ffffff&fg=333333&s=0 "6 \times 3 = magic[6+3] - magic[6-3]. \quad \quad \quad \quad (*)")
To find 
I’ve included enough magic numbers in the table so you can check that the scheme works for 
Can you reconstruct the values in the table, e.g. to find what ![magic[21]](http://s0.wp.com/latex.php?latex=magic%5B21%5D+&bg=ffffff&fg=333333&s=0 "magic[21]")
