login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A022144 Coordination sequence for root lattice B_2. 18
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 (list; graph; refs; listen; history; text; internal format)
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 (kitaev(AT)ms.uky.edu), 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

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.

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

S. 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.

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(binomial(2*n+1, 2*i)*z^i, i=0..n)-2*n*z*(1+z)^(n-1))/(1-z)^n. [Bacher et al.]

a(n) = (2n+1)^2 - (2n-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

EXAMPLE

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

MATHEMATICA

a=1; lst={a}; Do[b=n^2-a; AppendTo[lst, b]; a+=b, {n, 3, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, May 18 2009 *)

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.

Sequence in context: A277780 A044893 A185359 * A181390 A008590 A186544

Adjacent sequences:  A022141 A022142 A022143 * A022145 A022146 A022147

KEYWORD

nonn

AUTHOR

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 23 19:08 EST 2017. Contains 295128 sequences.