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
M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Séries 1) (1997), 1137-1142.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'institut Fourier, 49 no. 3 (1999), p. 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.
EXAMPLE
From Peter M. Chema, Mar 20 2016:
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
MATHEMATICA
CoefficientList[Series[(1 + 10 x + 7 x^2)/(1 - x)^2, {x, 0, 59}], x] (* Michael De Vlieger, Mar 21 2016 *)
PROG
(PARI) x='x+O('x^100); Vec((1+10*x+7*x^2)/(1-x)^2) \\ Altug Alkan, Mar 20 2016
CROSSREFS
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.
KEYWORD
nonn,easy
AUTHOR
Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)
STATUS
approved