Counting cumulative occupancy based on arrival date and length of stay

I need to work out the number of rooms occupied per day; the information I have is the date a person arrives and how long they stay. I suppose possibly it could be done by adding 1 to the daily total for a number of days equal to their length of stay? Really I have no idea how to do it!
So far I've been doing it with a chart, with 'room number' in the rows and 'date' in the columns - so if a person comes in on day 1 for 5 days I mark 5 cell in a row, then if a person comes in on day 2 for 10 days I mark 10 cells in the next row. Then I simply use COUNTA in the column to work out the total.
Can anyone either think of a way to automate the 'chart' method, or preferably some kind of formula to just give the numbers!

Thanks,

Harvey