Function that looks a lot like exponential, but isn't I'm looking for a continuous function f(x) with the following properties. I've been playing with exponentials, but that doesn't seem to be the answer, although my high school mathematics is a bit rusty, I must admit.


*

*$f(1) = 2$

*$f(2) = 6 = 2+4$

*$f(3) = 14 = 2+4+8$

*$f(4) = 30 = 2+4+8+16$

*And so on


I'm looking for a continuous function, so something with a meaningful answer for $f(2.5)$, which is less then $6 + \frac{14 - 6}{2} = 10$.
My simple high school math made me look at something like $f(x) = a^x$, but that doesn't seem to be the answer.
Any better ideas?

If you're computer-minded, it might seem natural to express these numbers in binary, since they're sums of powers of two!
Then you see that they're:
$$f(1)=10_2$$
$$f(2)=110_2$$
$$f(3)=1110_2$$
$$f(4)=11110_2$$
and it's immediately apparent that adding $10_2$ to these will give powers of two.
