A110907 Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.

1, 12, 50, 108, 194, 300, 434, 588, 770, 972, 1202, 1452, 1730, 2028, 2354, 2700, 3074, 3468, 3890, 4332, 4802, 5292, 5810, 6348, 6914, 7500, 8114, 8748, 9410, 10092, 10802, 11532, 12290, 13068, 13874, 14700, 15554, 16428, 17330, 18252, 19202

Offset

0,2

Comments

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

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

The sequence for the D_k lattice has the terms ((2*n+1)^k-(2*n-1)^k)/2, if k is even, and the terms ((2n+1)^k-(2*n-1)^k)/2+(-1)^n if k is odd (like here for k=3). The sequence for A_2 is A008458, for A_3 A010006, for A_4 the first differences of A083669. A_5 is 2+2*n^2*(25+44*n^2) if n>0, and 1 if n=0. - R. J. Mathar, Feb 09 2010

References

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

Links

Table of n, a(n) for n=0..40.

R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes arXiv:1002.3844 [math.GT], 2010.

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

Index entries for sequences related to f.c.c. lattice

Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Formula

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

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

a(n) = 1 + (-1)^n + 12*n^2, n>0.

G.f.: 1 - 2*x*(6 + 13*x + 4*x^2 + x^3)/((1+x)*(x-1)^3). (End)

Example

a(0) = 1: 000
a(1) = 12: +-1 +-1 0, where the 0 can be in any of the three coordinates
a(2) = 50: +-2 0 0 (6), +-2 +-1 +-1 (24), +-2 +-2 0 (12), +-2 +-2 +-2 (8).

Maple

A110907 := proc(n) a :=0 ; for x from -n to n do for y from -n to n do for z from -n to n do if type(x+y+z, 'even') then m := max( abs(x), abs(y), abs(z)) ; if m = n then a := a+1 ; end if; end if; end do ; end do ; end do ; a ; end proc: seq(A110907(n), n=0..40) ; # R. J. Mathar, Feb 03 2010

Mathematica

a[0] = 1; a[n_] := 1 + (-1)^n + 12*n^2;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 16 2017, after R. J. Mathar *)

Crossrefs

Cf. A117216, A022144, A010014, A175112 (D_5), A175114 (D_6).

Sequence in context: A335698 A081292 A052022 * A009937 A009932 A009933

Adjacent sequences: A110904 A110905 A110906 * A110908 A110909 A110910

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 15 2008

Extensions

I would like to get analogous sequences for A_2, A_4, A_5, ..., D_4 (see A117216), D_5, ..., E_6, E_7, E_8.

Extended by R. J. Mathar, Feb 03 2010

Removed the "conjectured" attribute from formulas - R. J. Mathar, Feb 27 2010

Status

approved