I tasked with solving the recurrence relation $a_n = 5a_{n-1} - 6a_{n-2}$ given the initial conditions $a_0 = 1, a_1 = 4$. I do not know how to begin here. However, I know that $$a_2 = 14,$$ $$a_3 = 46,$$ $$a_4 = 146,$$ $$a_5 = 454.$$ What is the pattern here?
Edit: I have explicit instructions to use generating functions, and then extract the coefficients using the GF.