The coordination sequence {S(n)} of a lattice or net gives the number
of nodes that are n bonds away from a given node. S(1) is the familiar
coordination number. Extending the work of O'Keeffe and others, we gi
ve explicit formulae for the coordination sequences of the root lattic
es A(d), D-d, E-6, E-7, E-8 and their duals. Proofs are given for many
of the formulae and for the fact that, in every case, S(n) is a polyn
omial in n, although some of the individual formulae are conjectural.
In the majority of cases, the set of nodes that are at most n bonds aw
ay from a given node form a polytopal cluster whose shape is the same
as that of the contact polytope for the lattice. It is also shown that
among all the Barlow packings in three dimensions the hexagonal close
packing has the greatest coordination sequence, and the face-centred
cubic lattice the smallest, as conjectured by O'Keeffe.