Solve the recurrence relation $a_n=3a_{n-1}-2a_{n-2}$ with initial conditions $a_0$=1 and $a_1$=3.

Find the generating function for the following series:

• $1, 1, 1, 1, \dots$
• $1, -1, 1, -1, 1, \dots$
• $1, 2, 3, 4, 5, \dots$
• $1, 3, 5, 7, \dots$
• $1, 4, 9, 16, \dots$

Find the generating function for the Fibonacci sequence.