login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A019557 Coordination sequence for G_2 lattice. 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This figurate number representing hexagram frames. Geometrically, the sequence is analogous to the expanding perimeters of triangles (A008486), squares (A008574), pentagons (A008706), and hexagons (A008458), but in relation to the hexagram lattice.  See the illustrations in the example below. - Peter M. Chema, Mar 28 2016

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

LINKS

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

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

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]

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 forming regular hexagrams, giving framed hexagram numbers:

                                                       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 the Star numbers A003154, Star of David matchstick numbers A045946, and Centered triangular hexagram numbers A270700.)

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

Cf. A003154, A045946, A270700, A000290, A008486, A008574, A008706, A008458.

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 * A173107 A277978 A131874

Adjacent sequences:  A019554 A019555 A019556 * A019558 A019559 A019560

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 26 23:07 EDT 2019. Contains 321566 sequences. (Running on oeis4.)