In
crystallography
Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
and the theory of infinite
vertex-transitive graph
In the mathematics, mathematical field of graph theory, an Graph automorphism, automorphism is a permutation of the Vertex (graph theory), vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-tr ...
s, the coordination sequence of a vertex
is an
integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
that counts how many vertices are at each possible distance from
. That is, it is a sequence
where each
is the number of vertices that are
steps away from
. If the graph is vertex-transitive, then the sequence is an
invariant of the graph that does not depend on the specific choice of
. Coordination sequences can also be defined for
sphere packing
In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...
s, by using either the
contact graph of the spheres or the
Delaunay triangulation
In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points; that is, each circumcircle has its gen ...
of their centers, but these two choices may give rise to different sequences.

As an example, in a
square grid, for each positive integer
, there are
grid points that are
steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence
in which, except for the initial value of one, each number is a multiple of four.
The concept was proposed by Georg O. Brunner and
Fritz Laves and later developed by
Michael O'Keefe
Michael O'Keefe (born Raymond Peter O'Keefe Jr.; April 24, 1955) is an American actor known for his roles as Danny Noonan in '' Caddyshack''; Ben Meechum in '' The Great Santini,'' for which he received a nomination for the Academy Award for Bes ...
. The coordination sequences of many low-dimensional lattices and
uniform tilings are known.
The coordination sequences of periodic structures are known to be
quasi-polynomial
In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi- ...
.
References
{{reflist, refs=
[{{cite OEIS, A008574, mode=cs2]
[{{citation
, last = Brunner , first = G. O.
, date = July 1979
, doi = 10.1016/0022-4596(79)90207-x
, issue = 1
, journal = Journal of Solid State Chemistry
, pages = 41–45
, title = The properties of coordination sequences and conclusions regarding the lowest possible density of zeolites
, volume = 29, bibcode = 1979JSSCh..29...41B
]
[{{citation
, last1 = Conway , first1 = J. H. , author1-link = John Horton Conway
, last2 = Sloane , first2 = N. J. A.
, date = November 1997
, doi = 10.1098/rspa.1997.0126
, issue = 1966
, journal = ]Proceedings of the Royal Society A
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905:
* Series A: for papers in physical sciences and mathematics.
* Series B: for papers in life s ...
, mr = 1480120
, pages = 2369–2389
, title = Low-dimensional lattices. VII. Coordination sequences
, volume = 453, bibcode = 1997RSPSA.453.2369C , s2cid = 120323174
[{{citation
, last1 = Goodman-Strauss
, first1 = C.
, last2 = Sloane
, first2 = N. J. A.
, author2-link = Neil Sloane
, arxiv = 1803.08530
, date = January 2019
, doi = 10.1107/s2053273318014481
, issue = 1
, journal = Acta Crystallographica Section A
, mr = 3896412
, pages = 121–134
, title = A coloring-book approach to finding coordination sequences
, url = https://neilsloane.com/doc/Cairo_final.pdf
, volume = 75
, pmid = 30575590
, s2cid = 4553572
, access-date = 2021-06-18
, archive-date = 2022-02-17
, archive-url = https://web.archive.org/web/20220217162848/http://neilsloane.com/doc/Cairo_final.pdf
, url-status = dead
]
[{{citation
, last = O'Keeffe , first = M.
, date = January 1995
, doi = 10.1524/zkri.1995.210.12.905
, issue = 12
, journal = Zeitschrift für Kristallographie – Crystalline Materials
, title = Coordination sequences for lattices
, volume = 210, pages = 905–908
, bibcode = 1995ZK....210..905O
]
[{{citation
, last1 = Shutov , first1 = Anton
, last2 = Maleev , first2 = Andrey
, date = 2020
, doi = 10.1515/zkri-2020-0002
, journal = Zeitschrift für Kristallographie – Crystalline Materials
, title = Coordination sequences for lattices
, volume = 235
, pages = 157–166
]
[{{citation
, last1 = Nakamura , first1 = Y.
, last2 = Sakamoto , first2 = R.
, last3 = Mase , first3 = T.
, last4 = Nakagawa , first4 = J.
, date = 2021
, doi = 10.1107/S2053273320016769
, journal = Acta Crystallogr.
, title = Coordination sequences of crystals are of quasi-polynomial type
, volume = A77
, issue = 2
, pages = 138–148
, pmid = 33646200
, pmc = 7941273
, bibcode = 2021AcCry..77..138N
]
[{{citation
, last = Kopczyński , first = Eryk
, date = 2022
, doi = 10.1107/S2053273322000262
, journal = Acta Crystallogr.
, title = Coordination sequences of periodic structures are rational via automata theory
, volume = A78
, issue = 2
, pages = 155–157
, pmid = 35230271
, arxiv = 2307.15803
]
Crystallography
Infinite graphs
Integer sequences