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A008264 Coordination sequence for tridymite, lonsdaleite, and wurtzite. 2
1, 4, 12, 25, 44, 67, 96, 130, 170, 214, 264, 319, 380, 445, 516, 592, 674, 760, 852, 949, 1052, 1159, 1272, 1390, 1514, 1642, 1776, 1915, 2060, 2209, 2364, 2524, 2690, 2860, 3036, 3217, 3404, 3595, 3792, 3994, 4202, 4414, 4632, 4855, 5084, 5317, 5556, 5800 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Lonsdaleite seems also to be called wurtzite. - N. J. A. Sloane, Apr 07 2018

REFERENCES

Inorganic Crystal Structure Database: Collection Code 29343

Michael O'Keeffe, Topological and geometrical characterization of sites in silicon carbide polytypes, Chemistry of Materials 3 (2) (1991), 332-335. (Eq. (2) gives an empirical formula for a(n). - N. J. A. Sloane, Apr 07 2018)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. L. Glasser, Symmetry properties of the wurtzite structure, Journal of Physics and Chemistry of Solids, 10(2-3) (1959), 229-241.

Ralf W. Grosse-Kunstleve, Zeolites, Frameworks, Coordination Sequences and Encyclopedia of Integer Sequences, 1996.

Ralf W. Grosse-Kunstleve, G. O. Brunner and N. J. A. Sloane, Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites, Acta Cryst., A52 (1996), 879-889.

Sean A. Irvine, Generating Functions for Coordination Sequences of Zeolites, after Grosse-Kunstleve, Brunner, and Sloane.

Michael O'Keeffe, N-dimensional diamond, sodalite and rare sphere packings, Acta Cryst. A 47 (1991), 749-753.

Reticular Chemistry Structure Resource (RCSR), The lon-b net.

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

FORMULA

a(4*m+k) = 42*m^2 + 21*k*m + [ 2, 4, 12, 25 ], 0 <= k < 4 (N. J. A. Sloane).

a(n) = 1 + (42*n^2 + (1 + (-1)^n)*(3 + 2*(-1)^((n - 1)*n/2)) + 6)/16 for n > 0, a(0) = 1. - Bruno Berselli, Jul 24 2013

G.f.: (1 + 2*x + 5*x^2 + 5*x^3 + 5*x^4 + 2*x^5 + x^6)/((1 - x)^3*(1 + x + x^2 + x^3)). - Bruno Berselli, Jul 24 2013

MATHEMATICA

a[n_] := (m = Quotient[n, 4]; k = Mod[n, 4]; 42*m^2 + 21*k*m + Switch[k, 0, 2, 1, 4, 2, 12, 3, 25]); a[0]=1; Table[a[n], {n, 0, 47}] (* Jean-Fran├žois Alcover, Oct 11 2012, from the first formula *)

Join[{1}, Table[1 + (42 n^2 + (1 + (-1)^n) (3 + 2 (-1)^((n - 1) n/2)) + 6)/16, {n, 50}]] (* Bruno Berselli, Jul 24 2013 *)

LinearRecurrence[{2, -1, 0, 1, -2, 1}, {1, 4, 12, 25, 44, 67, 96}, 20] (* Harvey P. Dale, Dec 27 2016 *)

PROG

(PARI) a(n)=if(n, 1+(42*n^2+(1+(-1)^n)*(3+2*(-1)^((n-1)*n/2))+6)/16, 1) \\ Charles R Greathouse IV, Feb 10 2017

CROSSREFS

Cf. A008524 for 4-D analog.

Sequence in context: A116668 A225254 A008186 * A000297 A078618 A304843

Adjacent sequences:  A008261 A008262 A008263 * A008265 A008266 A008267

KEYWORD

nonn,easy,nice

AUTHOR

Ralf W. Grosse-Kunstleve

STATUS

approved

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Last modified July 29 23:58 EDT 2021. Contains 346346 sequences. (Running on oeis4.)