A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

1, 40, 272, 888, 2080, 4040, 6960, 11032, 16448, 23400, 32080, 42680, 55392, 70408, 87920, 108120, 131200, 157352, 186768, 219640, 256160, 296520, 340912, 389528, 442560, 500200, 562640, 630072, 702688, 780680, 864240, 953560, 1048832, 1150248

Offset

0,2

Comments

This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.

The L_infinity norm of a vector is the largest component in absolute value.

Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Feb 05 2010

References

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

G. Nebe and N. J. A. Sloane, Home page for this lattice

Index entries for sequences related to D_4 lattice

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

From R. J. Mathar, Feb 03 2010, Feb 13 2010: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;

a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)

G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - Colin Barker, May 24 2012

Mathematica

CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4, {x, 0, 40}], x]  (* Vincenzo Librandi, Jun 27 2012 *)

Prog

(Magma) I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

Crossrefs

Cf. A110907, A175110.

Sequence in context: A247406 A229588 A334121 * A035099 A065255 A300920

Adjacent sequences: A117213 A117214 A117215 * A117217 A117218 A117219

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 15 2008

Extensions

a(2) corrected and sequence extended by R. J. Mathar, Feb 03 2010, Feb 13 2010

Status

approved