The CS is now routinely used to characterize crystallographic structures (Meier & Möck, 1979, Atlas of Zeolite Structure Types, 1992 & 1996, Fischer 1973 & 1974) and even higherdimensional sphere packings (Conway & Sloane, 1995, 1996). Other applications are to the determination of topological density, which can be obtained from the partial sums of terms in the CS. This density is correlated with other parameters such as lattice energy (Akporiaye & Price, 1989), the distribution of particular elements (Herrero, 1993), catalytic activity (Barthomeuf, 1993), and is useful for predicting of properties of synthetic zeolites (Brunner, 1979).
In an abstract sense, the CS describes the growth of a crystal, and was therefore initially called the "growth series" (Brunner & Laves, 1971). Previous investigations (Brunner, 1979, Herrero, 1994, Schumacher, 1994, O'Keeffe, 1991a) have shown that in some cases the terms in the CS increase quadratically with k, but up to now investigations were restricted to specific examples. In view of the increasing applications of the CS, up to 2000 terms have now been calculated for all approved zeolites topologies, as well as for all dense SiO_{2} polymorphs and 16 selected nontetrahedral structures, and the algebraic structure of these CS's has been analyzed.
The data for the twelve SiO_{2} polymorphs were taken from the Inorganic Crystal Structure Database (ICSD, 19861995). Table 1 lists the mineral names together with the ICSD collection codes.
For a comparison, 16 nontetrahedral structure types having many chemical representatives have also been investigated. Table 2 gives the search codes for the corresponding entries in the Gmelin Handbook (TYPIX, 1994).
The coordination numbers of the zeolites and the dense SiO_{2} polymorphs are always four, with the exception of the six interrupted zeolite frameworks (indicated by a dash preceding the threeletter code) which have a three or twofold coordination for one of the sites. For the nontetrahedral structures, Table 2 also gives the bond length limits and the resulting coordination numbers which were used in the calculations. For each of CaF_{2}, NiAs and W, two different parameters were considered, leading to two different coordination numbers.
The CS determination algorithm used here can be described as a node counting or coordination shell algorithm. The algorithm is started (with k = 0) by selecting an initial node. At the next step (k = 1), all nodes bonded to the initial node are determined. For k 2, all characteristics of the algorithm become evident: those nodes, which are bonded to the "new nodes of the previous step (k1)", but have not been counted before, are counted. This means that three sets of nodes for three topological distances (i.e. three coordination shells) have to be maintained: the middle (k1) nodes, whose bonds are followed to determine the next (k) nodes, and the previous (k2) nodes, to know which of the nodes bonded to the middle nodes have already been counted. The innermost shells with k < k_{next}2 are not needed and can be deleted since the nodes counted before are fully surrounded by shell k2. In this way, the memory required grows only quadratically with k, whereas other algorithms presented in the literature (Herrero, 1994) have a cubic growth rate.
Another important feature of the algorithm is that a "hashing" lookup technique was used to determine whether a newly generated node  a candidate for the next shell  was already in the middle or previous shell. This increased the speed of the program by about three orders of magnitude.
Both optimizations, with respect to memory and to speed, were necessary: without them it would not have been possible to compute the several hundred to several thousand CS terms that were required for the analysis. Computing times were a matter of hours on a highspeed workstation.
N_{k} = 5/2 k^{2} + 3/2 for k = 2 n + 1, n = 0,1,2,...
N_{k} = 5/2 k^{2} + 2 for k = 2 n + 2, n = 0,1,2,...
For ABW, the equations involve both k^{2} and k:
N_{k} = 19/9 k^{2} + 1/9 k + 16/9 for k = 3 n + 1, n = 0,1,2,...
N_{k} = 19/9 k^{2}  1/9 k + 16/9 for k = 3 n + 2, n = 0,1,2,...
N_{k} = 19/9 k^{2}  0 k + 2 for k = 3 n + 3, n = 0,1,2,...
In general, all coordination sequences investigated in this study can be described exactly by a periodic set of quadratic equations of the form:
(1)  
The number of equations, M, will be referred to as the period length. These equations hold for all k greater than or equal to some starting point k_{0}.
Following a definition of O'Keeffe (1991a), we define the "exact topological density" (TD) to be the mean of the a_{i} divided by the dimension N_{D} of the crystal space (which is 3 for all results presented here):
(2) 
As k increases, the effect of the linear and constant coefficients b_{i} and c_{i} on the CS decreases, and the quadratic coefficient a_{i} dominates. The error in the approximation
(3) 
(4) 
often provides a more concise description than the quadratic equations given in (1). The generating functions for the sequences considered in this study have the form:
(5) 
First, the GF's for some simple zeolites were obtained by using the Maple GFUN package (Waterloo Maple Software, 1994, Salvy & Zimmermann, 1994). However, more complex cases were beyond the capabilities of GFUN, and an alternative approach was used. Note that the Taylor series expansion of a generating function of the form (5) can be obtained by the simple recursive reconstruction algorithm shown in Fig. 1. This produces the coordination sequence from the sets IT and PL.
Conversely, given the set PL and a sufficient number of CS terms, the set IT can be obtained by multiplying the GF by the denominator of (5). This is accomplished by the recursive decomposition algorithm shown in Fig. 2.
Copy the CS terms to N_{0}...N_{k_max}

Figure 2 : Recursive decomposition algorithm 

(P2) Extra period lengths can be adjoined to the set PL, at the cost of enlarging the set IT. This corresponds to multiplying both the numerator and denominator of Eq. (5) by factors 1x^{n}.
(M1) Reduction of set IT with enlargement of set PL:
Starting with initial sets IT/PL, two "compact" sets IT_{c}/PL_{c} are obtained with the algorithm shown in Fig. 3,
For IT_{c}/PL_{c} the following relations hold for all CS's investigated:
(P4) Let IT_{c} and PL_{c} be the sets obtained with manipulation technique (M1). The starting point for the periodic set of quadratic equations can be taken to be the difference between the degrees of the numerator and denominator of (5), or in other words
(6) 
For example, for EUO T9 we have:
PL_{c} = { 36, 26, 24, 23, 22, 21, 17, 14, 10, 8, 5 }
(M2) Reversal of (M1)  Reduction of set PL with enlargement of set IT:
The number of elements of an initial set PL can be reduced with the algorithm shown in Fig. 4. Using this algorithm, it was possible to modify the generating functions for all the CS's investigated so that in every case exactly three PL elements were required.
Upper bounds on the subperiod lengths of the coefficients a_{i}, b_{i}, c_{i} can be established with a very simple algorithm. Based on IT_{c}/PL_{c}, about 3·LCM(PL_{c}) CS terms are computed. Next, the recursive decomposition algorithm is applied with PL = { LCM(PL_{c}), LCM(PL_{c}) }. With a linear period search in the resulting sequence, an upper bound pl_{a} on the subperiod length O(a_{i}) is obtained. The process is repeated with PL = { LCM(PL_{c}), pl_{a} } to obtain pl_{b}, and finally with PL = { pl_{a}, pl_{b} } to obtain pl_{c}.
For all sequences but those of GOO T3, JBW T1 & T2, TON T1 & T2, the Fe position of FeS_{2}Marcasite, and both position of CaF_{2}(2), the bounds pl_{a}, pl_{b} and pl_{c} are exactly equal to the subperiod lengths O(a_{i}), O(b_{i}) and O(c_{i}), respectively. For the exceptional cases (and of course also in general), the relations pl_{a} O(a_{i}), pl_{b} O(b_{i}) and pl_{c} O(c_{i}) hold. Moreover, for all CS's the relation pl_{a} pl_{b} pl_{c} holds, and O(a_{i}) O(b_{i}) O(c_{i}) is valid for all CS except those of GOO T3, JBW T1 & T2, the Fe position of FeS_{2}Marcasite, and both positions of CaF_{2}(2).
By application of the manipulation techniques (M2) and (M3), it was always possible to obtain O(PL) = 3 for all 390 threedimensional CS's that were investigated. It is conjectured that this holds for any threedimensional CS.
In simple cases, such as the f.c.c. or b.c.c. lattices, the elements of the set IT are positive integers. Indeed, it follows from the work of Stanley (1976 & 1980) that if certain conditions are satisfied (one of which is that the set of points in or on the kth coordination shell is convex), then the IT are necessarily positive. In the present investigation, however, many examples with negative IT elements were encountered. For example, the As position of NiAs(1) has IT = { 1, 4, 12, 10, 5, 2 }, PL = {2, 1, 1 }.
The period length of the second differences is four, as indicated by the parentheses. [Although it is not needed here, it is worth mentioning that Herschel's "circulator" notation provides a convenient terminology for describing such periodic sequences (Comtet, 1974, p. 109).]
The occurrence of a constant period in the second differences implies b_{i} = 0 for the entire (periodic) set of quadratic equations. For example, for ABW some b_{i} are different from zero (see above) and the numerical values of the second differences are not constant. However, their plot (Fig. 5) clearly reveals a periodicity, which can be used to estimate one period length for the recursive decomposition. In this case the period length is three. And indeed, the set of period lengths for ABW is { 3, 3, 1 }.
Figure 5 : Second differences of ABW 

(a) A set of n maximum period lengths PL_{max} = {pl(0)_{max}, pl(1)_{max}, ... , pl(n1)_{max}} is prescribed, and the first position is tested from 1 to pl(0)_{max}. At each pass of the loop, the recursive decomposition algorithm is applied to the CS, followed by a check for a sufficiently large sequence of zeros at the end of the resulting sequence. Also, a linear period seeking algorithm tries to recover a possibly last missing period length. In case of no success, pl(1) is also tested and a loop with two variables, but avoiding permutations, is executed. Unless a solution has been found, more and more loop positions are activated, until the entire set is depleted.
(b) A review of several known PL sets revealed that individual pl often occur in pairs. This observation and the exploitation of properties (P1) and (P2) led to the following strategy: Given a large number of CS terms, attempt to obtain a decomposition using PL = { m, m, m1, m1, ..., 2, 2, 1, 1 }.
Upon success, the initial set PL is subjected to manipulation technique (M1), in order to obtain IT_{c} and PL_{c}, the compact form of description of the particular CS.
ABW = ATN
LTA = RHO
CAN = AFG T2 = AFG T3 = LIO T2 = LIO T4 = LOS T2
GME = AFX T1
AFS T3 = BPH T3
EDI T1 = THO T1
ERI T1 = OFF T1
EAB T2 = OFF T2
Furthermore, the CS of one atom of Nd (the position at the origin) is equal to the CS of Mg, and one CS of NiAs(1) (again for the position at the origin) is equal to the CS of W(1).
Tables 35 list the results of the algebraic analysis. Column "S" gives the number of different CS's for the structure indicated in the first column. Except for AFG and LIO, which have two different sites with the same CS, this is also the number of crystallographically distinct positions. "O(IT)" and "O(PL)" designate the order of the set of initial terms and period lengths, respectively. For structures with more than one topologically distinct site, the range is given (e.g. EUO: 213233 initial terms). However, if there are only two values, these are separated by a comma instead of a hyphen (e.g. for EUO, O(PL) is either 10 or 11 for all ten sequences), and in some cases all sets have an equal number of members and only one value is necessary. "M" is the number of quadratic equations which make up the periodic set. "TD", the exact topological density defined by Eq. (2), is given exactly, as a rational fraction multiplied by N_{D} = 3 (TD · N_{D} = <a_{i}>) and, for better comparability, also as decimal number. "TD10" and "%" are defined by equations (7) and (8) below.
All the coordination sequences in this study have been added to the electronically accessible version of (Sloane & Plouffe, 1995) at the address sequences@research.att.com (Sloane, 1994). In this way, the CS can be used as a "fingerprint" to assist in the identification of a crystal structure.
A somewhat less special example is AFI. The parameters are a = 21/10, b = 0 for all values of k, and M = 10. The parameter c is palindromic about k = M/2 = 5. This means that c is the same for k = 1 and k = 9; it is also the same for the pairs 2&8, 3&7, and 4&6, respectively. For k = 10, the parameter c is 2. Such palindromic behavior about k = M/2 is frequently observed. An example in which M is odd is KFI with a = 12/7, b = 0 for all k, M = 7, where c is palindromic about k = 3.5 (thus c is the same for the pairs k = 1&6, 2&5, and 3&4, respectively). For the last term of the period (k = 7), c is again 2. Another kind of palindromic behavior may appear if b is not zero: the magnitudes of b and of c are the same for pairs of k as in the previous examples, but the signs of b may differ for the two members of a pair. An example is CAN.
Very often, the parameters a and b are the same for all values of k or show a much shorter period than c. An example is AFY: for site T1 b = 0 for all k, for atom T2, b has period 3 with the values 1/6, 1/6, 0 respectively, while c has a period of 30 if k is odd and 60 if k is even. In the examples mentioned so far, either b itself or the sum of the b's over the period is zero. This is not the case for either atom of milarite: all values of b are negative.
(7) 
Using Eq. (3) (TD is the same for all sites of a structure) leads to
(8) 
The rightmost column of Tables 35 (%) lists the percentage deviations of TD10_{norm} from the exact TD. Fig. 6 is a histogram of the distribution of the deviations. For the majority of the structures the deviation is well below 3%, but there are also some outliers. A deviation of more than five percent was found for CHI and CLO, two interrupted frameworks, for FAU, and for six of the nontetrahedral structures. These three zeolites represent relatively open and/or complex structures and one might conclude that more than ten steps are necessary in such cases in order to achieve a satisfactory convergence, but on the other hand very complex structures like PAU and MFI show a very good correlation between the exact topological density and TD10. In view of this, the additional effort necessary to obtain the exact TD seems justified, and previous work based on approximations to the exact value should perhaps be reconsidered.
Figure 6 : Histogram of TD/TD10 correlation (see Tables 35) 

In one dimension, the simplest periodic graph is a linear chain, for which the CS is 1, 2, 2, , with IT = { 1, 1 }, PL = { 1 }. In two dimensions, the hexagonal net 6^{3} (which occurs for example in the mineral biotite) has the CS 1, 3, 6, 9, 12, 15, , with IT = { 1, 1, 1 }, PL = { 1, 1 }. In three dimensions, as already mentioned, sodalite has IT = { 1, 1, 1, 1 }, PL = { 1, 1, 1 }. In a sense, these are the simplest coordination sequences in dimensions 1, 2 and 3.
O'Keeffe (1991b) has generalized these three structures to higher dimensions, defining "ndimensional sodalites" for all n. He also gives their coordination sequences for n 6. The generating functions of these sequences have been analyzed, and were found to continue the pattern of the first three: the set IT consists of n+1 1's, and PL of n 1's. It is reasonable to conjecture that this holds in general. In a forthcoming paper (Conway & Sloane, submitted) it will be shown that this conjecture is equivalent to the assertion that the points in or on the kth coordination shell of ndimensional sodalite are in onetoone correspondence with the ndimensional "centered tetrahedral" numbers.
However, the results are empirical, as there is no rigorous mathematical proof that a generating function of the form (5) must hold for the CS of a periodic structure. The applicability of Eq. (5) has been verified for certain one, two and threedimensional periodic topologies. In the case of one and two dimensions, the justification is straightforward, but already in three dimensions there are difficulties. But this is a relatively minor point. Once the generating function has been discovered, watching its Taylor series expansion match the CS for hundreds or even thousands of terms carries complete conviction!
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^{1} The structure type code ROG has been discredited (see IZA Structure Commission Report, 1994) and is included only for completeness. 
^{2}
The Atlas of Zeolite Structure Types is available on the WorldWideWeb at
http://www.izasc.ethz.ch/IZASC/
Acta Crystallographica Section A, Volume A52 (1996), pages 879889.
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