You probably won't be able to solve this

A given function on the natural numbers satisfies

f(0) = 0

and

{f(2)-f(1), f(3)-f(2), f(4)-f(3), f(5)-f(4), ...}
=
{1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, ...}

is there a "nice" formula for f(n)?

Not well posed. f(1) is arbitrary.