Counting of [network] nodes

The number of nodes reached by going out to network distance r (with r>1) from any node in the networks on page 477 is (a) 4r-4, (b) 3/2 r^^{2} - 3/2 r + 1, and (c)

First[Select[4/9 r^^{3} + 2/3 r^^{2} + {2, 5/3, 5/3} r - {10/9, 1, -4/9}, IntegerQ]]

In any trivalent network, the quantity f[r] obtained by adding up the numbers of nodes reached by going distance r from each node must satisfy f[0]=n and f[1]=3n, where n is the total number of nodes in the network. In addition, the limit of f[r] for large r must be n^^{2}. The values of f[r] for all other r will depend on the pattern of connections in the network.