Determine the generating function A(x)=sumn=0∞anxnA(x) = sum_{n=0}^{\infty} a_n x^nA(x)=sumn=0∞anxn for the recurrence an=3an−1+4an−2a_n = 3a_{n-1} + 4a_{n-2}an=3an−1+4an−2 with a0=0,a1=1a_0 = 0, a_1 = 1a0=0,a1=1.
A(x)=x1−3x−4x2A(x) = \frac{x}{1-3x-4x^2}A(x)=1−3x−4x2x
A(x)=11−3x−4x2A(x) = \frac{1}{1-3x-4x^2}A(x)=1−3x−4x21
A(x)=x(1−4x)(1+x)A(x) = \frac{x}{(1-4x)(1+x)}A(x)=(1−4x)(1+x)x
A(x)=x1+3x−4x2A(x) = \frac{x}{1+3x-4x^2}A(x)=1+3x−4x2x