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

%I A319980 #32 Jan 19 2020 18:13:17
%S A319980 1,3,4,8,12,11,18,19,18,28,26,25,38,33,32,48,40,39,58,47,46,68,54,53,
%T A319980 78,61,60,88,68,67,98,75,74,108,82,81,118,89,88,128,96,95,138,103,102,
%U A319980 148,110,109,158,117,116,168,124,123,178,131,130,188,138,137,198,145,144,208,152,151
%N A319980 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 3.9^2.
%C A319980 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.
%C A319980 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.
%D A319980 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.
%H A319980 N. J. A. Sloane, <a href="/A319980/b319980.txt">Table of n, a(n) for n = 0..10000</a>
%H A319980 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</a>.
%H A319980 Jean-Guillaume Eon, <a href="https://doi.org/10.1107/S0108767301016609">Algebraic determination of generating functions for coordination sequences in crystal structures</a>, Acta Cryst. A58 (2002), 47-53. See Section 8.
%H A319980 N. J. A. Sloane, <a href="/A066393/a066393.jpg">A portion of the (9^3, 3.9^2) net</a>
%H A319980 N. J. A. Sloane, <a href="/A319980/a319980.png">Trunks and branches structure used to find the coordination sequence</a> (Blue = trunks, red = branches, green = twigs. The two thick red lines are the special branches.)
%H A319980 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A319980 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.
%p A319980 A319980 := proc(n) local t1,t2,t3,k,r;
%p A319980 t1:=[1,3,4,8,12,11,18,19];
%p A319980 t2:=[20,14,14,20,14,14];
%p A319980 t3:=[-2,5,4,8,12,11];
%p A319980 if n <= 7 then t1[n+1]
%p A319980 else k:=floor(n/6); r:=n-6*k;
%p A319980 t2[r+1]*k+t3[r+1]; fi; end;
%p A319980 [seq(A319980(n),n=0..128)];
%t A319980 Join[{1, 3}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {4, 8, 12, 11, 18, 19}, 64]] (* _Jean-François Alcover_, Feb 07 2019 *)
%Y A319980 Cf. A066393.
%K A319980 nonn,easy
%O A319980 0,2
%A A319980 _N. J. A. Sloane_, Oct 13 2018