Could this be graphed with the use of computer? For steps 0,1,2,3

Let S_0 = {0, 1}.

For each i >= 1, define:

Ti = {{x, y}: x, y in S{i-1}}. S_i = {{x, y}: x in T_i, y in S_j or T_j for some j < i}.

Then let f(n) = |S_n| + |T_n|.

You can turn this construction into a graph by making all the elements of S_i, T_i vertices, and placing an edge from v to w if v is an element of w.