The Cantor ternary function,
*F* : [0,1] -> [0,1]
is defined as follows:

- Express
*x*in base 3. If possible, use no 1s. (This makes a difference only if the expansion ends in 022222... = 100000... or 200000... = 122222...) - Replace the first 1 with a 2 and everything after it with 0.
- Replace all 2s with 1s.
- Interpret the result as a binary number. The result is
*F*(*x*).

A graph of the Cantor function is shown below.

This month's problem is to evaluate

For *n* = 1,2, and 3. Can you find a formula for arbitrary
*n*?