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A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.

%I A066393 #28 Jun 12 2025 00:48:41
%S A066393 1,3,6,6,12,15,12,21,24,18,30,33,24,39,42,30,48,51,36,57,60,42,66,69,
%T A066393 48,75,78,54,84,87,60,93,96,66,102,105,72,111,114,78,120,123,84,129,
%U A066393 132,90,138,141,96,147,150,102,156,159,108,165,168,114
%N A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.
%C A066393 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 9^3. See also A319980.
%H A066393 Muniru A Asiru, <a href="/A066393/b066393.txt">Table of n, a(n) for n = 0..300</a>
%H A066393 Jean-Guillaume Eon, <a href="https://doi.org/10.1006/jssc.1998.7754">Geometrical relationships between nets mapped on isomorphic quotient graphs: examples</a>, Journal of Solid State Chemistry 138.1 (1998): 55-65. See Fig. 1.
%H A066393 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 A066393 N. J. A. Sloane, <a href="/A066393/a066393.jpg">A portion of the (9^3, 3.9^2) net</a>
%F A066393 G.f.: (1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2.
%F A066393 a(n) = (3*n + lcm(n,3))/2, for n>=1. - _Ridouane Oudra_, Jan 22 2021
%F A066393 a(n) = 3*A186101(n), for n>=1. - _Ridouane Oudra_, Jun 11 2025
%p A066393 seq(coeftayl((1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2, x = 0, k), k=0..60); # _Muniru A Asiru_, Feb 13 2018
%Y A066393 Cf. A319980, A109044, A186101.
%K A066393 nonn
%O A066393 0,2
%A A066393 _N. J. A. Sloane_, Dec 24 2001