Prozentrechner

Here's a structured approach to solving the functional equation:

Given:

f(m+n)=f(m)+f(n)−2f(mn+m+n+1)+m2+n2f(m + n) = f(m) + f(n) - 2f(mn + m + n + 1) + m^2 + n^2f(m+n)=f(m)+f(n)−2f(mn+m+n+1)+m2+n2

for all nonnegative integers m,nm, nm,n, and f(1)=0f(1) = 0f(1)=0.

Base Case: f(1)=0f(1) = 0f(1)=0 is given.

Plugging in Small Values:

Set m=0m = 0m=0, n=0n = 0n=0, and find f(0)f(0)f(0).

Set m=1m = 1m=1, n=0n = 0n=0, and analyze.

Continue deriving a pattern for f(n)f(n)f(n).

Inductive or Substitution Approach: Solve iteratively for f(123)f(123)f(123).