Closed formula for two variable recurrence
I do not have a strong mathematical background, but I am afraid that recursive functions cannot always been written in closed form.

ex1.

f(n) = sum([1,2,..,n]) can be written as
f(1) = 1
f(n) = n + f(n-1)

Of course f(n) = n*(n+1)/2

ex2.

f(n) = prod([1,2,3,4,..n]) = n!
f(1) = 1
f(n) = n * f(n-1)

No known closed formula :( (I don't even know how to prove something like this)

In your case, I find it extremely difficult because your recursion makes use of an if statement under the min() function.