Question

Express the solution to the following counting problem in terms of a difference
equation, then use a generating function to solve the difference equation to give an
explicit solution:
Bob is threading beads onto a wire to make a decorative keyring. He has an
unlimited supply of 2 cm long beads in a choice of red, orange, yellow, green, blue,
or purple, as well as an unlimited supply of 1 cm long black beads. If the wire is n
cm long, how many different sequences of beads are possible?