Here is one way.

How many ways can we spend 2, 5, and 10 dollar bills using at most 2010 dollars? The 1 dollar bills are used to make up any difference. Include 0 as valid way to spend 2,5,10.

Fix the number of 10 dollar bills as 0. Then we spend at most 2010 dollars with 2 and 5. Fix # of 5s at 0, then # ways is 2010/2+1. Fix # of 5's at 1 then # ways is floor(2005/2)+1. Write the sum.

Now fix number of 10 dollar bills as 1. We spend at most 2000 with 2 and 5. Fix # 5 at 0, get 2000/2+1. Etc. Write sum.

Compute sum of sums.