for range of cell = n, calculate sum of cell + n-1 previous cell

I have a product that is purchased for a period of time ranging from 1 to 12 month. So I have a cell with the value that can be anything between 1 and 12.

I also have a table in a separate sheet that has 12 columns, each with a new instance of that product.

I want to track existing contracts (eg a contract that is running for 3 month would still show up in column 2 and 3 but then disappear in column 4.

I could create an array of 12 cells by 12 cells to account for each possibility but that seems to be a drag and I suspect I can do this via formula.

Given values
A in column 1
B in column 2
C in column 3
D in column 4
...
L in column 12

The array looks like this:

	1	2	3	4	..	12
1	A	B	C	D	..	L
2	A	A+B	B+C	C+D	...	K+L
3	A	A+B	A+B+C	B+C+D	...	J+K+L
4	A	A+B	A+B+C	A+B+C+D	...	I+J+K+L
..
12	A	A+B	A+B+C	A+B+C+D	...	A+B+C+D+E+F+G+H+I+J+K+L

Based on the choice I have on the first column, it would return the appropriate sum that would populate the field "existing customer contracts"

Any ideas?

PS: Here's an example of what I want to do: example.xlsx