How does the density of states (DoS) of Jellium scale with the number of electrons for constant electronic density?

I'm trying to understand the behavior of the density of states when we grow a system, such as $$C\_n H\_{2n+2}.$$ Since this is probably too complicated, I am interested in jellium first. Is there any description of this density of states as a function of the number of particles and energy somewhere? Ideally, this would look like $$f(E, N),$$ modelled as complexity estimates.


Thanks!


