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 writeup, 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):
Level 
Cost 
1 
1 
2 
2 
3 
3 
4 
4 
5 
5 
6 
10 
7 
30 
8 
60 
9 
90 
10 
120 
11 
160 
12 
200 
13 
250 
14 
310 
15 
390 
16 
485 
17 
610 
18 
760 
19 
950 
20 
1185 
21 
1480 
22 
1760 
23 
2070 
24 
2435 
25 
2865 
26 
3370 
27 
3965 
28 
4665 
29 
5490 
30 
6455 
31 
7595 
32 
8490 
33 
9435 
34 
10,485 
35 
11,650 
36 
12,945 
37 
14,380 
38 
15,980 
39 
17,755 
40 
19,725 
41 
21,920 
42 
24,355 
43 
27,060 
44 
30,065 
45 
33,405 
46 
37,115 
47 
41,240 
48 
45,825 
49 
50,915 
50 
56,570 
51 
62,855 
52 
69,840 
53 
77,600 
54 
86,220 
55 
95,800 
56 
106,445 
57 
118,275 
58 
131,415 
59 
146,015 
60 
162,240 
61 
180,265 
62 
200,295 
63 
222,550 
64 
247,280 
65 
274,755 
66 
305,280 
67 
339,200 
68 
376,890 
69 
418,765 
70 
465,295 
71 
516,995 
72 
574,440 
73 
638,225 
74 
709,185 
75 
787,985 
76 
875,535 
77 
972,820 
78 
1,080,910 
79 
1,201,010 
80 
1,334,455 
81 
1,482,725 
82 
1,647,475 
83 
1,830,525 
84 
2,033,915 
85 
2,259,910 
86 
2,511,010 
87 
2,790,010 
88 
3,100,010 
89 
3,444,455 
90 
3,827,170 
91 
4,252,415 
92 
4,724,905 
93 
5,249,890 
94 
5,833,210 
95 
6,481,346 
96 
7,201,496 
97 
8,001,660 
98 
8,890,735 
99 
9,878596 
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 OnLine 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 highdegree 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:
Level 
Cost 
Common Ratio (T_{n}/T_{(n1)}) 
1 
1 

2 
2 
2.00000000 
3 
3 
1.50000000 
4 
4 
1.33333333 
5 
5 
1.25000000 
6 
10 
2.00000000 
7 
30 
3.00000000 
8 
60 
2.00000000 
9 
90 
1.50000000 
10 
120 
1.33333333 
11 
160 
1.33333333 
12 
200 
1.25000000 
13 
250 
1.25000000 
14 
310 
1.24000000 
15 
390 
1.25806452 
16 
485 
1.24358974 
17 
610 
1.25773196 
18 
760 
1.24590164 
19 
950 
1.25000000 
20 
1185 
1.24736842 
21 
1480 
1.24894515 
22 
1760 
1.18918919 
23 
2070 
1.17613636 
24 
2435 
1.17632850 
25 
2865 
1.17659138 
26 
3370 
1.17626527 
27 
3965 
1.17655786 
28 
4665 
1.17654477 
29 
5490 
1.17684887 
30 
6455 
1.17577413 
31 
7595 
1.17660728 
32 
8490 
1.11784068 
33 
9435 
1.11130742 
34 
10,485 
1.11128776 
35 
11,650 
1.11111111 
36 
12,945 
1.11115880 
37 
14,380 
1.11085361 
38 
15,980 
1.11126565 
39 
17,755 
1.11107635 
40 
19,725 
1.11095466 
41 
21,920 
1.11128010 
42 
24,355 
1.11108577 
43 
27,060 
1.11106549 
44 
30,065 
1.11104952 
45 
33,405 
1.11109263 
46 
37,115 
1.11106122 
47 
41,240 
1.11114105 
48 
45,825 
1.11117847 
49 
50,915 
1.11107474 
50 
56,570 
1.11106747 
51 
62,855 
1.11110129 
52 
69,840 
1.11112879 
53 
77,600 
1.11111111 
(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:
 14: linear
 59: possibly exponential with a rounding error
 1021: exponential with a common ratio of 5/4
 2231: exponential with a common ratio of 1/0.85
 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):
 14: X_{n} = n
 59: X_{n} = 5*ROUND((1/5)*(3/2)^(n1),0)
 1021: X_{n} = 5*ROUND((51/15)*(5/4)^(n1),0)
 2231: X_{n} = 5*ROUND(58/5*(1/0.85)^(n1),0)
 31?: X_{n} = 5*ROUND(11665/36/5*(1/0.9)^(n1),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.