Did you notice the pattern in these numbers. The first two are 1 and 2, and after that, each term is the sum of the previous two.
3 = 1 + 2, 5 = 2 + 3, 8 = 3 + 5, ...
This sequence is called the Fibonacci sequence after an Italian mathematician Leonordo Fibonacci (1170, 1250).
There is an explanation of why his sequence gives the number of possible combinations in your stair problem in a note that Denis wrote in 1995.