Find the generating function A(x)=∑n=0∞anxnA(x) = \sum_{n=0}^{\infty} a_n x^nA(x)=∑n=0∞anxn for the recurrence an=6an−1−9an−2a_n = 6a_{n-1} - 9a_{n-2}an=6an−1−9an−2 with a0=1,a1=6a_0 = 1, a_1 = 6a0=1,a1=6.
A(x)=11−6x+9x2A(x) = \frac{1}{1-6x+9x^2}A(x)=1−6x+9x21
A(x)=11−6xA(x) = \frac{1}{1-6x}A(x)=1−6x1
A(x)=1−3x1−6x+9x2A(x) = \frac{1-3x}{1-6x+9x^2}A(x)=1−6x+9x21−3x
A(x)=1+3x1−6x+9x2A(x) = \frac{1+3x}{1-6x+9x^2}A(x)=1−6x+9x21+3x