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A319980 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 3.9^2. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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).

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.

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-Fran├žois Alcover, Feb 07 2019 *)

CROSSREFS

Cf. A066393.

Sequence in context: A263768 A050316 A161538 * A170885 A080646 A187579

Adjacent sequences:  A319977 A319978 A319979 * A319981 A319982 A319983

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Oct 13 2018

STATUS

approved

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Last modified August 4 01:37 EDT 2021. Contains 346441 sequences. (Running on oeis4.)