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

Author

N. J. A. Sloane, Apr 15 2008

Keywords

• nonn
• easy

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

Examples

			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).
		

References

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

Links

• 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).

Crossrefs

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

Programs

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 *)

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)

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