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%I #64 Mar 05 2024 09:07:28
%S 1,12,30,48,66,84,102,120,138,156,174,192,210,228,246,264,282,300,318,
%T 336,354,372,390,408,426,444,462,480,498,516,534,552,570,588,606,624,
%U 642,660,678,696,714,732,750,768,786,804,822,840,858,876,894,912,930,948,966,984,1002,1020,1038,1056
%N Coordination sequence for G_2 lattice.
%C Also, coordination sequence of Dual(3.12.12) tiling with respect to a 12-valent node. - _N. J. A. Sloane_, Jan 22 2018
%C For n > 1, also the number of minimum vertex colorings of the n-Andrásfai graph. - _Eric W. Weisstein_, Mar 03 2024
%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://dx.doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Séries 1) (1997), 1137-1142.
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://www.numdam.org/item?id=AIF_1999__49_3_727_0">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, Annales de l'institut Fourier, 49 no. 3 (1999), p. 727-762.
%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>
%H N. J. A. Sloane, <a href="/A019557/a019557.png">Illustration of layers 0,1,2 in the graph of the Dual(3.12.12) tiling</a>. Centered at a 12-valent node. Note that some of the blue edges are not part of the underlying graph.
%H N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndrasfaiGraph.html">Andrásfai Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MinimumVertexColoring.html">Minimum Vertex Coloring</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 18*n-6, n >= 1.
%F G.f.: (1 + 10*x + 7*x^2)/(1-x)^2.
%e From _Peter M. Chema_, Mar 20 2016:
%e Illustration of initial terms:
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%e 1 12 30 48
%e Compare to A003154, A045946, and A270700. (End)
%t CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* _Michael De Vlieger_, Mar 21 2016 *)
%o (PARI) x='x+O('x^100); Vec((1+10*x+7*x^2)/(1-x)^2) \\ _Altug Alkan_, Mar 20 2016
%Y Cf. A003154, A045946, A270700, A000290, A008486, A008574, A008706, A008458.
%Y For partial sums see A082040.
%Y List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
%K nonn,easy
%O 0,2
%A Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)
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