<s>[INST] Closed form for $(a_n)$ such that $a_{n+2} = \frac{a_{n+1}a_n}{6a_n - 9a_{n+1}}$ with $a_1=1$, $a_2=9$ $$a_1 = 1; a_2 = 9; a_{n+2} = \frac{a_{n+1}a_n}{6a_n - 9a_{n+1}}$$
I need to find non-recurring formula for $a_n$. Is there any good way to do this? The only one comes to mind is to guess the formula and then prove it using mathematical induction. 
Thanks in advance! 
I've got the result and it looks like this: $a_n = \frac{-3*2^{n-1} + 2^{2n - 1} + 1}{3}$ but I really don't like this way and would love to know how to solve this properly. 
 [/INST] As in Did's answer, but simpler to handle:
$$
b_{n + 2} = 6 b_{n + 1} - 9 b_n \qquad b_1 = 1, b_2 = 1/9
$$
Define $B(z) = \sum_{n \ge 0} b_{n + 1} z^n$. By the properties of generating functions:
$$
\frac{B(z) - b_1 - b_2 z}{z^2} = 6 \frac{B(z) - b_1}{z} - 9 B(z)
$$
From here:
$$
B(z) = \frac{9 - 53 z}{9 - 54 z + 81 z^2}
     = \frac{53}{27} \frac{1}{1 - 3 z} - \frac{26}{27} \frac{1}{(1 - 3 z)^2}
$$
By the expansion (a form of the binomial theorem):
$$
(1 - u)^{-m} = \sum_{n \ge 0} \binom{n + m - 1}{m - 1} z^n
$$
we get:
$$
b_{n + 1} = \frac{53}{27} 3^n - \frac{26}{27} \binom{n + 1}{1} 3^n
          = (27 - 26 n) \cdot 3^{n - 3}
$$
so that finally:
$$
a_n = \frac{1}{(53 - 26 n) \cdot 3^{n - 4}}
$$
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