Let m, k be fixed positive integers. Determining the generating function for the number of tilings of an m × n rectangle by k × 1 rectangles is a longstanding open problem to which the answer is only known in certain special cases. We give an explicit formula for this generating function in the case where m < 2k. This result is used to obtain the generating function for the number of tilings of an m × n × k box with k × k × 1 bricks.