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Coordination sequence for root lattice B_2.
39

%I #86 Sep 08 2022 08:44:45

%S 1,8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144,152,160,

%T 168,176,184,192,200,208,216,224,232,240,248,256,264,272,280,288,296,

%U 304,312,320,328,336,344,352,360

%N Coordination sequence for root lattice B_2.

%C 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

%C Number of points of L_infinity norm n in the simple square lattice Z^2. - _N. J. A. Sloane_, Apr 15 2008

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

%C 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

%C 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

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

%H Vincenzo Librandi, <a href="/A022144/b022144.txt">Table of n, a(n) for n = 0..10000</a>

%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, Zeit. f. Kristallographie, 212 (1997), 253-256.

%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://dx.doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

%H Tom Karzes, <a href="/A250122/a250122.html">Tiling Coordination Sequences</a>

%H Sergey Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.

%H Sergey Kitaev, <a href="https://web.archive.org/web/20130625171839/http://www.ms.uky.edu/~math/MAreport/4-ser.ps">On multi-avoidance of right angled numbered polyomino patterns</a>, University of Kentucky Research Reports (2004).

%H Joan Serra-Sagrista, <a href="http://dx.doi.org/10.1016/S0020-0190(00)00119-8">Enumeration of lattice points in l_1 norm</a>, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.

%H N. J. A. Sloane, <a href="/A296368/a296368_2.png">Overview of coordination sequences of Laves tilings</a> [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

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

%F 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.]

%F 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

%F a(n) = 0^n + 8*n. - _Vincenzo Librandi_, Aug 21 2011

%F G.f.: 1 + 8*x/(1-x)^2. - _R. J. Mathar_, Feb 16 2018

%F Sum_{i=0..n} a(i) = (2*n+1)^2 = A016754(n). - _Chunqing Liu_, Jan 12 2020

%F E.g.f.: 1 + 8*x*exp(x). - _Stefano Spezia_, Apr 05 2021

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

%t Join[{1}, LinearRecurrence[{2, -1}, {8, 16}, 50]] (* _Jean-François Alcover_, Jan 07 2019 *)

%o (Magma) [0^n+8*n: n in [0..50] ]; // _Vincenzo Librandi_, Aug 21 2011

%Y Apart from initial term, the same as A008590.

%Y Cf. A234275.

%Y For partial sums see A016754.

%Y 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.

%K nonn,easy

%O 0,2

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