This is a continuation of [this](https://math.stackexchange.com/questions/1408121/finding-a-function-such-that-n-2-2-fn-1-leq-fn) post where I tried to find a function $f(n)$ that would satisfy the induction step of an inductive argument and it was shown that such function does not exist.

Trying to fix the problem I've come up with a stronger inductive argument that now requires finding a more elaborate function. 

More precisely I would like to find a function $f(n,k)$ such that the following relation would hold for any $0 \leq p \leq (n-2-l)/2$ and $l \leq k$, $l \leq (n-2)$

$$p + f(n-1, k-l+p) \leq f(n,k).$$ 

$k \geq 0$ can be bounded to anything as long as the induction step still works out.

> 
> Is there a function $f(n,k)$ that satisfies the above inequality such that $$f(n,0) \leq \frac{n^2-6n+6}{6}.$$
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Notice that if we take $f(n,k) = n^2-k$, then this gives a solution, but I need the function $f(n,0)$ to be smaller.