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A110907 Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

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

R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes arXiv:1002.3844 [R. J. Mathar, Feb 27 2010]

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

CROSSREFS

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

Sequence in context: A029586 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

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Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.