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A008458 Coordination sequence for hexagonal lattice. 16
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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.CO/0508136.

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (Abstract, pdf, ps).

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

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

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.: F(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

EXAMPLE

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)

MAPLE

[ seq(6*n, n=0..45) ]; # (except for initial term)

MATHEMATICA

Join[{1}, 6*Range[60]] (* Harvey P. Dale, Jul 21 2013 *)

PROG

(PARI) a(n) = 6*n+(!n);

(MAGMA) [0^n+6*n: n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

(Maxima) makelist(if n=0 then 1 else 6*n, n, 0, 30); /* Martin Ettl, Nov 12 2012 */

CROSSREFS

Essentially the same as A008588.

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

Sequence in context: A126798 A175130 * A008588 A078596 A187389 A085129

Adjacent sequences:  A008455 A008456 A008457 * A008459 A008460 A008461

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 24 13:10 EDT 2014. Contains 240983 sequences.