searching for $f(x)$ when knowing $f(2^{2x})$ I have a function that works for powers of 2
This is only for INTEGERS
is there a way to calculate any integer x?
$$f(2^{2x})=\frac{4^x+2}3.$$
$f(x)$=?
here are the first 100 values of f(x)
1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, 6, 6, 8, 8, \ 14, 14, 16, 16, 14, 14, 16, 16, 22, 22, 24, 24, 22, 22, 24, 24, 30, \ 30, 32, 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, \ 32, 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, 32, \ 32, 30, 30, 32, 32, 22, 22, 24, 24, 22, 22, 24, 24, 30, 30, 32, 32, \ 30, 30, 32, 32, 54, 54, 56, 56, 54 is there a pattern to calculate any f(x)?
thanx

Note that $4^x=2^{2x}$, so you can simply swap them both for $x$ and get
$$
f(x)=\frac{x+2}{3}
$$
This new description is only valid for positive real $x$, and even then only if the original description was valid for any real $x$. If the original expression was only valid for integer $x$, for instance, then it would probably be better to leave it unchanged.
