A008264 Coordination sequence for tridymite, lonsdaleite, and wurtzite.

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

Offset: 0

Author

Ralf W. Grosse-Kunstleve

Keywords

• nonn
• easy
• nice

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.
• Michael O'Keeffe, Topological and geometrical characterization of sites in silicon carbide polytypes, Chemistry of Materials 3 (2) (1991), 332-335.
• Reticular Chemistry Structure Resource (RCSR), The lon net (lonsdaleite) and The lon-b net (wurtzite).
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

Cf. A008524 for 4-D analog, A008253 for diamond.

Cf. A217511 for theta series.

Programs

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-François 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 *)

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

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