Let: $g: \mathbb{N} \rightarrow \mathbb{R^+}$ a function with $g(1) = c$ and $g(x) = 2g(\frac{x}{2}) + cx^2$ and assume that $x$ is a power of 2. I want to finde a closed form with telescoping.
The first part should be $cn$, if i'am right. But i don't see what happens with $cx^2$
With $h(n):=\frac 1cg(2^n)$, we are given $h(0)=1$ and $h(n)=2h(n-1)+2^{2n}$. Writing down the binary expansion of the first few values, you will arrive at the conjecture that $h(n)=2^{2n+1}-2^n$. This can then reaily proved by induction: $h(0)=1=2^1-2^0$, and if $h(n-1)=2^{2(n-1)+1}-2^{n-1}$, then $$h(n)=2h(n-1)+2^{2n}=2^{2(n-1)+2}-2^{n-1+1}+2^{2n}=2^{2n+1}-2^n.$$