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

v

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

v

. That is, it is a sequence

n_0, n_1, n_2,\dots

where each

n_i

is the number of vertices that are

i

steps away from

v

. If the graph is vertex-transitive, then the sequence is an invariant of the graph that does not depend on the specific choice of

v

. 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. Distance countours in a square grid

As an example, in a square grid, for each positive integer

i

, there are

4i

grid points that are

i

steps away from the origin. Therefore, the coordination sequence of the square grid is the sequence

1,4,8,12,16,20,\dots\ .

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