Recursion describes the interaction of percentage-modifying 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.

Nunu Nunu

  • Denote Nunu Nunu's base health as a, bonus health (without counting the above multipliers) as b, total health percentage increase due to Consume.png Consume as x, total health percentage increase due to runes as y, bonus health percentage increase due to Juggernaut item.png Cinderhulk 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.

Ryze Ryze

Define Ryze Ryze's raw AP as the ability power without counting Seraph's Embrace item.png Seraph's Embrace's Awe and ability power multipliers (e.g. Rabadon's Deathcap item.png Rabadon's Deathcap's passive or Infernal Drake Infernal Drake's buff). Denote it as a. Denote the ability power multiplier as x. Define raw mana as the mana without counting Arcane Mastery.png Arcane Mastery. Denote it as b.

The resultant AP and resultant mana can be calculated as follows:

Define two sequences \pagecolor{Black}\color{White}A = a_n and \pagecolor{Black}\color{White}B = b_n where \pagecolor{Black}\color{White}n = 0, 1, 2, \ldots.

Let \pagecolor{Black}\color{White}a_0 = ax, b_0 = b. For \pagecolor{Black}\color{White}n \geq 1, \pagecolor{Black}\color{White}a_n = 0.03 b_{n-1} x, b_n = 0.0005 a_{n-1} b.

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 formulas:

Resultant AP: \pagecolor{Black}\color{White}\frac{ax + 0.03 bx}{1 - 0.000015 bx} Resultant mana: \pagecolor{Black}\color{White}\frac{b + 0.0005 abx}{1 - 0.000015 bx}

Unyielding mastery 2016.png Unyielding and Fearless mastery 2017.png Fearless

  • Denote the base armor as a, bonus armor (without counting the above multipliers) as b, bonus armor percentage increase due to Unyielding mastery 2016.png Unyielding as x, total armor percentage increase due to Fearless mastery 2017.png Fearless as y, and flat bonus from Fearless mastery 2017.png Fearless 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.

Vladimir Vladimir

PENDING FOR TEST