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, 1258000, 1372280

Offset: 0

Author

N. J. A. Sloane, Apr 15 2008

Keywords

• nonn
• easy
• changed

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
• Gabriele 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).

Crossrefs

Cf. A035878, A105374, A110907, A144965, A175110.

Programs

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

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 *)

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

E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - Elmo R. Oliveira, Aug 18 2025

Extensions

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