<s>[INST] Recurrence $k$-th pattern by substitution The recurrence is $T(n) = 9 T(\frac{n}{3}) + {n^2}$.
Simplified to ${3^2} T(\frac{n}{3}) + {n^2}$. Then
$$\begin{align}
T(n)&= 3^4T(\frac{n}{3^2}) + 4n\\
&= 3^6T(\frac{n}{3^3}) + 10n\\
&= 3^8T(\frac{n}{3^4}) + 28 n.
\end{align}$$
I am trying to find the $k$-th pattern of this recurrence.
 [/INST] In my opinion something is wrong in your approach.
If $T(n)={3^2} T(\frac{n}{3}) + {n^2}$, then $T(\frac{n}{3^k})={3^2} T(\frac{n}{3^{k+1}}) + (\frac{n}{3^k})^2$ and
$$\begin{align}T(n)&={3^2} T(\frac{n}{3}) + {n^2}={3^2} \left({3^2} T(\frac{n}{3^2}) + \frac{n^2}{3^2}\right) + n^2\\
&={3^4} T(\frac{n}{3^2})  + 2n^2
={3^4} \left({3^2} T(\frac{n}{3^3}) + \frac{n^2}{3^4}\right) + 2n^2\\
&={3^6} T(\frac{n}{3^3})  + 3n^2=\dots={9^k} T(\frac{n}{3^k})  + kn^2.\\
\end{align}$$
P.S. Do you know the Master Theorem?
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