Recursion describes the interaction of percentage-modifying stacking of champion statistics. To have a recursion you need 2 or more multipliers, each one increasing the other's effect, infinitely but with diminishing returns.

General

All formulas of different interactions can be found by making an equation system of all the variables. For example considering Master YiSquare Master Yi's Wuju Style Wuju Style and Guinsoo's Rageblade item Guinsoo's Rageblade interaction we have this:

Wuju Style increases total attack damage, so we have: WujuBonus = (base + bonus + GuinsooBonus)  × WujuStyle's multiplier

Guinsoo instead increases bonus attack damage: GuinsooBonus = (bonus + WujuBonus)  × Guinsoo's multiplier

By resolving the system you find the values of WujuBonus and GuinsooBonus based on your base and bonus attack damage. The total attack damage is going to be = base + bonus + WujuBonus + GuinsooBonus which is:

$ \pagecolor{black}\color{White} {{BaseAD \times (1 + Wuju Multiplier) + BonusAD \times (1 + Wuju Multiplier) \times (1 + Guinsoo Multiplier)} \over {1 - (Wuju Multiplier) \times (Guinsoo Multiplier)}} $

The above part of the formula is the simplest since you multiply the base or bonus attack damage by their multipliers, while the below part changes depending on how many recursion interactions are there. With two it's 1 - (Wuju Multiplier) × (Guinsoo Multiplier). With three (e.g Wuju + Guinsoo + Dragon Slayer) it's: $ \pagecolor{black}\color{White} 1 - (Wuju) \times (Guinsoo) - (Wuju) \times (Dragon) - (Guinsoo) \times (Dragon) - 2 \times (Wuju) \times (Guinsoo) \times (Dragon) $

NunuSquare Nunu

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

RyzeSquare Ryze

Define RyzeSquare Ryze's raw AP as the ability power without counting Seraph's Embrace item Seraph's Embrace's Awe and ability power multipliers (e.g. Rabadon's Deathcap item Rabadon's Deathcap's passive or Infernal DrakeSquare 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 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} $

Other recursions