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

Author

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

Keywords

• nonn
• easy

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 i1Sergey 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

Examples

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

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).

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.

Programs

Magma

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

Mathematica

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

Python

def A022144(n): return n<<3 if n else 1 # Chai Wah Wu, Mar 11 2025

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