If you are one of those folks that doesn't like math, this may be a good time to stick your fingers in your ears and hum. Things are about to get very nerdy!
D&D has experience tables with a progression that simulates an exponential function but instead of producing a gentle curve, it is all jerky and crude. Every version of D&D has been like this from the beginning. The progression tables may change but the numbers are arbitrarily selected instead of being calculated. Experience may go up by 1000 per level for a few levels, then by 1500 for a few levels and then by 2000 for a few levels. It's functional but not pretty. Being a bit of a math nerd, I want it to be pretty.
A Very Useful Family of Functions
Years ago I noticed that if you make a list of numbers from one to ten and then make a list next to them of the sum of the numbers in the first list down to that point, you would create a natural, simple progression. If you then make a third list that takes the sum of the numbers in the second list, you could create another simple progression related to the first but climbing more rapidly.
More correctly, these two functions would be "n * (n+1)/2" and "n * (n+1)/2 * (n+2)/3". I'm sure that I'm not the first person to discover this family of functions. If anyone happens to know the name for this family, I'd love to hear about it!
The Same Number of Encounters Every Level
In D&D v4.0, if each encounter is between five characters and five monsters all of the same level, the experience for the encounter is roughly one tenth of that needed to gain a level. That's easy to accomplish. You just need to match your experience table with the monster experience value table. For some reason they did this through 20th level (with one exception) but deviated on the rest of the levels to 30.
I like the idea of having the experience from ten encounters exactly match the experience needed to gain a level. And it's very easy to tie this to the progression above.
A Matter of Scale
I'd stop here but there is one other issue I want to address. There is no reason that I can see to deal with unnecessarily large numbers. After a point, some players have trouble relating to numbers on a scale larger than they are used to dealing with. Therefore, I want to drop the scale by a factor of ten. As long as the experience table and the monster experience value table are both in proportion to each other, the scale of the numbers doesn't matter.
In one case you would have ten encounters of 1000 exp each taking you to your goal of 10,000. In the second case you would have ten encounters of 100 exp each taking you to your goal of 1,000. It works out to exactly the same thing. It just uses numbers on a smaller scale to do so. Even though the numbers are smaller, characters will level at exactly the same rate.
So here is the final result. The first column is the level. The second column is the monster experience value table (Using the first function times 10). The third column is the experience table (using the second function times 100).
Test It Out
A character at the exact beginning of level 20 would have 133,000 experience and need to reach 154,000 experience to gain level 21. That's a difference of 21,000 experience. A single level 20 monster yields 2,100 experience so a group of five level 20 monsters would give 10,500 experience. Split that between five level 20 adventurers, and they would get 2,100 experience each for one encounter or 21,000 experience each for ten similar encounters.
So What Did All That Get Us?
Honestly, the difference is practically nil. We turned an ugly pseudo-function into a more elegant true function. There were a few levels where ten encounters wouldn't quite gain a level. Now ten encounters is exactly a level all the time. And we reduced the scale of the numbers involved to something people can more readily relate to. Not exactly world-shattering stuff. But I and my fellow math nerds may sleep a bit more easily knowing that there is now some actual math behind those tables. :)