A319980 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 3.9^2.

1, 3, 4, 8, 12, 11, 18, 19, 18, 28, 26, 25, 38, 33, 32, 48, 40, 39, 58, 47, 46, 68, 54, 53, 78, 61, 60, 88, 68, 67, 98, 75, 74, 108, 82, 81, 118, 89, 88, 128, 96, 95, 138, 103, 102, 148, 110, 109, 158, 117, 116, 168, 124, 123, 178, 131, 130, 188, 138, 137, 198, 145, 144, 208, 152, 151

Offset: 0

Author

N. J. A. Sloane, Oct 13 2018

Keywords

• nonn
• easy

Comments

This net may be regarded as a tiling of the plane by 9-gons and triangles. There are two kinds of vertices: (a) 9^3 vertices, where three 9-gons meet, and (b) 3.9^2 vertices, where a triangle and two 9-gons meet. The present sequence is the coordination sequence with respect to a vertex of type 3.9^2. See also A066393.

The coordination sequence was found using the "coloring book" method. The link below shws the trunsks and branches structure. The calculations are very similar to those used for the 3.12.12 uniform tiling in the CGS-NJAS paper.

References

• Eon, Jean-Guillaume. "Geometrical relationships between nets mapped on isomorphic quotient graphs: examples." Journal of Solid State Chemistry 138.1 (1998): 55-65. See Fig. 1.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
• Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See Section 8.
• N. J. A. Sloane, A portion of the (9^3, 3.9^2) net
• N. J. A. Sloane, Trunks and branches structure used to find the coordination sequence (Blue = trunks, red = branches, green = twigs. The two thick red lines are the special branches.)
• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A066393.

Programs

Maple

A319980 := proc(n) local t1,t2,t3,k,r;
t1:=[1,3,4,8,12,11,18,19];
t2:=[20,14,14,20,14,14];
t3:=[-2,5,4,8,12,11];
if n <= 7 then t1[n+1]
else k:=floor(n/6); r:=n-6*k;
t2[r+1]*k+t3[r+1]; fi; end;
[seq(A319980(n),n=0..128)];

Mathematica

Join[{1, 3}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {4, 8, 12, 11, 18, 19}, 64]] (* Jean-François Alcover, Feb 07 2019 *)

Formula

G.f.: (1 + 3*x + 4*x^2 + 6*x^3 + 6*x^4 + 3*x^5 + 3*x^6 - 2*x^7)/(1 - x^3)^2.