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A110907
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Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.
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6
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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; internal format)
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OFFSET
| 0,2
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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. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 09 2010]
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REFERENCES
| J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.
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LINKS
| 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 (mathar(AT)strw.leidenuniv.nl), Feb 27 2010]
Index entries for sequences related to f.c.c. lattice
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1)
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FORMULA
| 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). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 03 2010]
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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).
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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) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 03 2010]
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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
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Apr 15 2008
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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 (mathar(AT)strw.leidenuniv.nl), Feb 03 2010
Removed the "conjectured" attribute from formulas - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 27 2010
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