164121

I think you could find all the multiples of the number of the rectangles, for instance if there were 100 rectangles in the stick you could find the multiples-in this case 1, 2, 4, 5, 10, 20, 25, 50- you would then find there multiple pairs e.g 25 and 4 or 5 and 20, and would then see how many factor pairs there are and how big the number of a factor pair is. This would give you lots of numbers which would need to be added up and them would be your total amount of rectangles there could be. I think this method would give at least an estimate of how many possible rectangles there could be in the counting stick.