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# Recursion

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Recursion is a method of stacking of champion statistics. It involves 2 or more multipliers, each of which interacts with the bonus statistics contributed by all others. This leads to an infinite sum on the statistic.

## Examples

• Define  raw AP as the ability power without counting 's Awe and ability power multipliers (e.g. 's passive or 's buff). Denote it as a.
• Denote the ability power multiplier as x.
• Define raw mana as the mana without counting . Denote it as b.
• The resultant AP and resultant mana can be calculated as follows:
• Define two sequences A = {an} and B = {bn}, where n = 0, 1, 2, ...
• a0 = ax, b0 = b
• For n ≥ 1, an = 0.03bn-1x, bn = 0.0005an-1b
• The resultant AP is the sum of all terms in A, while the resultant mana is the sum of all terms in B.
• This gives the formula
```   Resultant AP = (ax + 0.03bx) / (1 - 0.000015bx),
Resultant mana = (b + 0.0005abx) / (1 - 0.000015bx)
```
• Denote base health as a, bonus health (without counting the above multipliers) as b, total health percentage increase due to as x, total health percentage increase due to runes as y, bonus health percentage increase due to as z.
• z = 15% = 0.15
• The resultant health can be calculated as follows:
• Define two sequences S = {sn} and T = {tn}, where n = 0, 1, 2, ...
• s0 = a + b, t0 = 0
• s1 = (a + b)y + bz, t1 = (a + b)x
• For n ≥ 2, sn = tn-1[(1 + y)(1 + z) - 1], tn = sn-1x
• The resultant health is the sum of all terms in S and T.
• This gives the formula
```   Resultant health = [a(1 + y) + b(1 + y + z)] (1 + x) / [1 - x(yz + y + z)]
```
• To calculate the interaction between any two of them, just put the remaining modifier (x, y or z) to 0.
• Denote the base armor as a, bonus armor (without counting the above multipliers) as b, bonus armor percentage increase due to as x, total armor percentage increase due to as y, and flat bonus from as c
• y = 10% = 0.1
• The resultant armor can be calculated as follows:
• Define two sequences S = {sn} and T = {tn}, where n = 0, 1, 2, ...
• s0 = a + b + c, t0 = 0
• s1 = (a + b)y, t1 = (b + c)x
• For n ≥ 2, sn = tn-1y, tn = sn-1x
• The resultant armor is the sum of all terms in S and T.
• This gives the formula
```  Resultant armor = [a(1 + y) + b(1 + x)(1 + y) + c(1 + x)] / (1 - xy)
```
• Same works for magic resistance.

PENDING FOR TEST

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