Let $F(m,n)$ be defined recursively for non-negative integers $m$ and $n$ according to the following rules:
- $F(0,n) = 0$ for all $n$,
- $F(m,n) = F(n,m)$ for all $m$ and $n$, and
- if $n\ge m$, then $F(m,n) = F(m,n-m) + \lfloor (m-1)^2/4\rfloor$.
What is a closed form expression for $F(m,n)$?
I've computed the first few values of $F(m,n)$, but cannot find the pattern:
$$ \begin{array} {r|r r r r r r r r r r} m\backslash n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\ \hline 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3\\ 4 & 0 & 0 & 1 & 2 & 2 & 2 & 3 & 4 & 4 & 4\\ 5 & 0 & 0 & 1 & 2 & 4 & 4 & 4 & 5 & 6 & 8\\ 6 & 0 & 0 & 2 & 2 & 4 & 6 & 6 & 6 & 8 & 8\\ 7 & 0 & 0 & 2 & 3 & 4 & 6 & 9 & 9 & 9 & 11\\ 8 & 0 & 0 & 2 & 4 & 5 & 6 & 9 & 12 & 12 & 12\\ 9 & 0 & 0 & 3 & 4 & 6 & 8 & 9 & 12 & 16 & 16\\ 10 & 0 & 0 & 3 & 4 & 8 & 8 & 11 & 12 & 16 & 20 \end{array}$$