A019557 Coordination sequence for G_2 lattice.

1, 12, 30, 48, 66, 84, 102, 120, 138, 156, 174, 192, 210, 228, 246, 264, 282, 300, 318, 336, 354, 372, 390, 408, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 642, 660, 678, 696, 714, 732, 750, 768, 786, 804, 822, 840, 858, 876, 894, 912, 930, 948, 966, 984, 1002, 1020, 1038, 1056

Offset

0,2

Comments

Also, coordination sequence of Dual(3.12.12) tiling with respect to a 12-valent node. - N. J. A. Sloane, Jan 22 2018

For n > 1, also the number of minimum vertex colorings of the n-Andrásfai graph. - Eric W. Weisstein, Mar 03 2024

Links

Table of n, a(n) for n=0..59.

Michael Baake and Uwe Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), pp. 253-256

Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Séries 1) (1997), pp. 1137-1142.

Roland Bacher, Pierre de la Harpe, and Boris Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'institut Fourier, 49 no. 3 (1999), pp. 727-762.

Tom Karzes, Tiling Coordination Sequences.

N. J. A. Sloane, Illustration of layers 0,1,2 in the graph of the Dual(3.12.12) tiling. Centered at a 12-valent node. Note that some of the blue edges are not part of the underlying graph.

N. J. A. Sloane, Overview of coordination sequences of Laves tilings. [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

Eric Weisstein's World of Mathematics, Andrásfai Graph.

Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 18*n - 6, n >= 1.

G.f.: (1 + 10*x + 7*x^2)/(1-x)^2.

From Elmo R. Oliveira, Apr 04 2025: (Start)

E.g.f.: 6*exp(x)*(3*x - 1) + 7.

a(n) = 6*A016789(n-1) for n >= 1.

a(n) = 2*a(n-1) - a(n-2) for n >= 3. (End)

Example

From Peter M. Chema, Mar 20 2016: (Start)
Illustration of initial terms:
                                                       o
                                                      o o
                                    o                o   o
                                   o o        o o o o o o o o o o
                  o           o o o o o o o    o   o       o   o
               o o o o         o o     o o      o o         o o
     o          o   o           o       o        o           o
               o o o o         o o     o o      o o         o o
                  o           o o o o o o o    o   o       o   o
                                   o o        o o o o o o o o o o
                                    o                o   o
                                                      o o
                                                       o
     1           12                30                 48
Compare to A003154, A045946, and A270700. (End)

Mathematica

CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* Michael De Vlieger, Mar 21 2016 *)

Prog

(PARI) my(x='x+O('x^100)); Vec((1+10*x+7*x^2)/(1-x)^2) \\ Altug Alkan, Mar 20 2016

Crossrefs

Cf. A000290, A003154, A008706, A016789, A045946, A270700.

For partial sums see A082040.

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.

Sequence in context: A256620 A125582 A031107 * A396980 A173107 A277978

Adjacent sequences: A019554 A019555 A019556 * A019558 A019559 A019560

Keyword

nonn,easy

Author

Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)

Status

approved