A022144 Coordination sequence for root lattice B_2.

1, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360

Offset

0,2

Comments

Equivalently, the coordination sequence for a point of degree 8 in the tiling of the Euclidean plane by right triangles (with angles Pi/2, Pi/4, Pi/4). These triangles are fundamental regions for the Coxeter group (2,4,4). In the notation of Conway et al. 2008 this is the tiling *442. The coordination sequence for a point of degree 4 is given by A234275. - N. J. A. Sloane, Dec 28 2015

Number of points of L_infinity norm n in the simple square lattice Z^2. - N. J. A. Sloane, Apr 15 2008

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 24 ).

Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004

These numbers correspond to the number of primes in the shells of a prime spiral. In a(2) there are 8 primes surrounding 2 in a prime spiral. - Enoch Haga, Apr 06 2000

References

J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 191.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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 (Series 1) (1997), 1137-1142.

Tom Karzes, Tiling Coordination Sequences

Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.

Sergey Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).

Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

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

Formula

a(n) = [x^(2*n)] ((1 + x)/(1 - x))^2.

G.f. for coordination sequence of B_n lattice: Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i - 2*n*z*(1+z)^(n-1)/(1-z)^n. [Bacher et al.]

a(n) = (2*n+1)^2 - (2*n-1)^2. Binomial transform of [1, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Dec 27 2007

a(n) = 0^n + 8*n. - Vincenzo Librandi, Aug 21 2011

G.f.: 1 + 8*x/(1-x)^2. - R. J. Mathar, Feb 16 2018

Sum_{i=0..n} a(i) = (2*n+1)^2 = A016754(n). - Chunqing Liu, Jan 12 2020

E.g.f.: 1 + 8*x*exp(x). - Stefano Spezia, Apr 05 2021

Example

1 + 8*x + 16*x^2 + 24*x^3 + 32*x^4 + 40*x^5 + 48*x^6 + 56*x^7 + ...

Mathematica

Join[{1}, LinearRecurrence[{2, -1}, {8, 16}, 50]] (* Jean-François Alcover, Jan 07 2019 *)

Prog

(Magma) [0^n+8*n: n in [0..50] ]; // Vincenzo Librandi, Aug 21 2011

Crossrefs

Apart from initial term, the same as A008590.

Cf. A234275.

For partial sums see A016754.

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: A044893 A185359 A365886 * A181390 A008590 A186544

Adjacent sequences: A022141 A022142 A022143 * A022145 A022146 A022147

Keyword

nonn,easy

Author

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

Status

approved