Generating function for non-linear recurrence relation

Question

What will be the generating function for recurrence relation of following type

$f(n)$ = $f(n/2)$ + $n^2$

please help

Answer 1

Hint

Suppose than $$f(n)=a+b n+c n^2$$ So $$f(n)-f\left(\frac{n}{2}\right)-n^2=0 \implies \frac{b }{2}n+\frac{3 c }{4}n^2-n^2=0$$ Make all coefficients equal to $0$.

If you are not convinced, try $$f(n)=a+b n+c n^2+dn^3$$ $$f(n)-f\left(\frac{n}{2}\right)-n^2=0 \implies \frac{b }{2}n+\frac{3 c }{4}n^2-n^2+\frac {7 d} 8n^3=0$$ Make all coefficients equal to $0$.