Hi, I tried to solve below word problem
Express the solution of the following problem in terms of a difference equation, then use a generating function to solve the difference equation to give an explicite solution.
Every week my grandmother knits either a hat that uses one ball of wool, or a scarf that uses two balls of wool. If the hats are all black and the scarves can be blue, red, or green, yellow, purple or orange, how many different ways are there for her to use up n balls of wool? (We assume that the order in which she knits the items matters.)
I know that the difference equation is of the form . Now, the initial conditiones are? , as using one ball of wool you get only 1 hat, and when using 2 balls of wool you get either 2 hats or one scarf. I also know the form of the generating function but, I am not sure about the initial conditiones, I would appreciate any help.
Thanks in advance.