Let's Get Nerdy: Plane Attribute Costs

I'm writing a huge section on Planes at the moment (soon to be published), and a statement I saw on Facebook got me thinking.  The post was discussion of max resource production, and the statement was as follows:

"vampire is better than f117, just more expensive"

This was in reference to Development Zone garrison slot, where I've been using F117 with Christmas Frost gears for a long time, and I'm reasonably happy with my development numbers.  As many of us did, I started out with Vampire garrisoned in development zone until I got an F117 and was amazed by how fast I was able to max it out.

I'll delve into which plane is better to garrison in Development Zone in my Planes write-up, but the question that got me thinking is: how expensive is it to max out a Vampire?  The only way to know this is to predict the cost in Air Force Tactical Points for Attribute steps beyond what I've already unlocked (and thus know the costs for).  Therefore, I want to write a formula that will allow me to calculate the cost in AFTP for a given Attribute step.

Here are the steps and their associated costs that I know so far (from my own planes):

I first tried to use an online pattern finder, but I was quickly foiled.  No section of the series of numbers was matched by any of the online pattern finders I checked.  (I also tried The On-Line Encyclopedia of Integer Sequences® (OEIS®), but came up blank there too). More math was needed.

I started out by trying to classify the type of equation I was dealing with.  It's obviously not linear, so I looked to polynomials.  An online search led me to a paper at the Carnegie Mellon University Mathematical Sciences department titled "Finding a formula for a sequence of numbers".  This paper described a process for determining if a given sequence of numbers could be solved by a high-degree polynomial equation, but following the process led me to determine that my desired formula was not a polynomial sequence.  The suggestion of that paper was to investigate an exponential sequence.

I had already checked to see if this was a logarithmic sequence, and had determined that it was not.  Another Google search took me to a lesson on CK12.org titled "Geometric Sequences and Exponential Functions."  Now I was getting somewhere.

The process on that lesson suggested that I investigate the Common Ratio, or the ratio between successive terms in the sequence.

This is what I found:

(That 1/0.9 pattern continues throughout the data that I've collected, I've just cut it off to shorten this already very long entry.)

I can see five distinct sections that have patterns:

1. 1-4: linear
2. 5-9: possibly exponential with a rounding error
3. 10-21: exponential with a common ratio of 5/4
4. 22-31: exponential with a common ratio of 1/0.85
5. 31-?: exponential with a common ratio of 1/0.9

The final values are also rounded to the nearest number divisible by 5.  Here are my resulting Google Sheets formulas (where n is the step, or level of the next Attribute, and X_n is the cost of that Attribute):

1. 1-4: X_n = n
2. 5-9: X_n = 5*ROUND((1/5)*(3/2)^(n-1),0)
3. 10-21: X_n = 5*ROUND((51/15)*(5/4)^(n-1),0)
4. 22-31: X_n = 5*ROUND(58/5*(1/0.85)^(n-1),0)
5. 31-?: X_n = 5*ROUND(11665/36/5*(1/0.9)^(n-1),0)

These formulas are not perfect, but they successfully predict the known costs of an Attribute for each step up to level 99, with a margin of error of about 0.028%.

The question is, how high does this formula take us?  Some AMS systems reach a plateau for costs (troop enhancement via spray, I believe), so does this formula go to ∞, or does it peak out at some maximum cost per step?  If we assume it applies to ∞ (and this is a big assumption), we can build a spreadsheet that gives us the cost for each step.

Getting back to our Vampire - the Vampire attributes have two tracks: the right side is the development track, where most (but not all) of the attributes we want for maximum resource production are.  The left side is the deployment track, which is mostly (but not completely) comprised of battle attributes.  We can now count the number of steps required to max out a side, then sum up the costs for each step to find out how many Air Force Tactical Points it takes to max each side.

To max out the left (deployment) it requires 150 steps (we get this by adding the number of steps for each Attribute).  If we assume the formula above carries on to ∞, my calculations show that this equates to 21.3 Billion Air Force Tactical Points to max out a Vampire for Attack purposes.  This is compared to the 109.8 Million required to max out the F117.

To max out the right (development) side of a Vampire it requires 162 steps.  Again, if we assume the formula above carries to ∞, my calculations suggest that it takes 75.4 Billion Air Force Tactical Points to max out a Vampire for Development purposes.

What if we want to max out the Vampire for both sides?  We can't simply add 21.3B + 75.4B.  Instead, you have to calculate the sum of all costs for 312 steps.  This calculates to a whopping  495,911,656,025,378,000, or 495.9 Quadrillion Air Force Tactical Points.

The good news?  You can use the Air Force Tactical Points +50% Battlefield Buff to cut all of these numbers in half!

Now, I must admit, I don't know if my assumption that the formula applies to ∞ is correct, and that it doesn't max out at some steady cost instead.  I did hear on Facebook that one person was actually able to max their Vampire and it took them 3 years, and this suggests to me that there might be a maximum cost per step at some level higher than 98.  If anyone from AMS (or anyone else for that matter) wants to check my math and let me know anything I'm missing I'm all ears.