puzzle creation - Burning ropes as timers - How many time intervals can be measured?

^{Note: This is a follow-up question to Burning ropes as timers. The following question and its answers may contain spoilers.}

To sum up the puzzle, these are the rules (slightly modified):

• You have some number N of ropes, each with the following property.


• When you light one end of a rope, the fire will reach the other end after exactly one hour.


• You start the puzzle by lighting one or more ends of the ropes.


• You may light or extinguish any end of a rope after that.


• You can only light or extinguish a rope once another rope has completely burned out.



• You may only light the end of a rope, since lighting anywhere else is inaccurate.


• You may not organize the ropes so that they light each other or themselves.


• You must accurately measure some interval of time, between any two distinct times.

The question is, how many intervals of time (excluding 0) is it possible to measure with N ropes?

For example, when N = 1, the answer is 2.

• 1 hour (by burning the length of the rope)


• 1/2 hour (by lighting both ends of the rope at once)