Find the generating function A(x)=∑n=0∞anxnA(x) = \sum_{n=0}^{\infty} a_n x^nA(x)=∑n=0∞anxn for the sequence an=4an−1−3an−2a_n = 4a_{n-1} - 3a_{n-2}an=4an−1−3an−2 with a0=1,a1=5a_0 = 1, a_1 = 5a0=1,a1=5.
1+x1−4x+3x2\frac{1+x}{1-4x+3x^2}1−4x+3x21+x
11−4x+3x2\frac{1}{1-4x+3x^2}1−4x+3x21
1−x1−4x+3x2\frac{1-x}{1-4x+3x^2}1−4x+3x21−x
1+5x1−4x+3x2\frac{1+5x}{1-4x+3x^2}1−4x+3x21+5x