Solve recurrence by generating functions

Question

Find non-recurrent expression for the following sequence: $a_0=a_1=1\;\; 5a_{n+2}=4a_{n+1}-a_n$

The formula I got for the respective generating function: $$5(A(x)-1-x)=4x(A(x)-1)-x^2A(x)$$ $$A(x)=\frac{x+5}{5-4x+x^2}$$ The denominator is irreducible, so I can't get further by partial fractions expansion.

Answer 1

If you do not want to use complex numbers, you can start by completing the square, giving $$ x^2 - 4x + 5 = (x - 2)^2 + 1 $$ Hence \begin{align*} A(x) &= \frac{x+5}{(x-2)^2 + 1}\\ &= (x+5) \sum_{k=0}^\infty (-1)^k(x-2)^{2k}\\ \end{align*} Now expand using the binomial theorem.