A008458 Coordination sequence for hexagonal lattice.

1, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. It is also the planar net 3.3.3.3.3.3.

Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_6].

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 20 ).

Also the Engel expansion of exp^(1/6); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002

Numbers k such that k+floor(k/2) | k*floor(k/2). - Wesley Ivan Hurt, Dec 01 2020

Examples

			From _Omar E. Pol_, Aug 20 2011: (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
. 1             o o o       o       o       o           o
.       6                    o o o o         o         o
.                 12                          o o o o o
.                               18
.                                                 24
(End)
G.f. = 1 + 6*x + 12*x^2 + 18*x^3 + 24*x^4 + 30*x^5 + 36*x^6 + 42*x^7 + 48*x^8 + 54*x^9 + ...
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
• Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also on arXiv, arXiv:1803.08530 [math.CO], 2018-2019.
• Rostislav Grigorchuk and Cosmas Kravaris, On the growth of the wallpaper groups, arXiv:2012.13661 [math.GR], 2020. See section 4.1 p. 19.
• Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons, Mathematics Magazine, 50 (1977), 227-247.
• Tom Karzes, Tiling Coordination Sequences
• G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
• Reticular Chemistry Structure Resource, hxl
• N. J. A. Sloane, The uniform planar nets and their A-numbers [Annotated scanned figure from Gruenbaum and Shephard (1977)]
• 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]
• William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
• William A. Stein, The modular forms database
• Index entries for sequences related to A2 = hexagonal = triangular lattice
• Index entries for linear recurrences with constant coefficients, signature (2,-1).
• Index entries for coordination sequences

Crossrefs

Essentially the same as A008588.

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574(4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579(3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529(3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).

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.

Cf. A032528. - Omar E. Pol, Aug 20 2011

Cf. A048477 (binomial Transf.)

Programs

Magma

[0^n+6*n: n in [0..60] ]; // Vincenzo Librandi, Aug 21 2011

Maple

1, seq(6*n, n=1..65);

Mathematica

Join[{1},6*Range[60]] (* Harvey P. Dale, Jul 21 2013 *)
a[ n_] := Boole[n == 0] + 6 n; (* Michael Somos, May 21 2015 *)

Maxima

makelist(if n=0 then 1 else 6*n,n,0,65); /* Martin Ettl, Nov 12 2012 */

PARI

{a(n) = 6*n + (!n)};

SageMath

[6*n+int(n==0) for n in range(66)] # G. C. Greubel, May 25 2023

Formula

G.f.: (1 + 4*x + x^2)/(1 - x)^2.

a(n) = A003215(n) - A003215(n-1), n > 0.

Equals binomial transform of [1, 5, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 08 2008

G.f.: Hypergeometric2F1([3,-2], [1], -x/(1-x)). - Paul Barry, Sep 18 2008

a(n) = 0^n + 6*n. - Vincenzo Librandi, Aug 21 2011

n*a(1) + (n-1)*a(2) + (n-2)*a(3) + ... + 2*a(n-1) + a(n) = n^3. - Warren Breslow, Oct 28 2013

E.g.f.: 1 + 6*x*exp(x). - Stefano Spezia, Jun 26 2022