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%I #28 Nov 16 2017 06:22:38
%S 1,12,50,108,194,300,434,588,770,972,1202,1452,1730,2028,2354,2700,
%T 3074,3468,3890,4332,4802,5292,5810,6348,6914,7500,8114,8748,9410,
%U 10092,10802,11532,12290,13068,13874,14700,15554,16428,17330,18252,19202
%N Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.
%C This lattice consists of all points (x,y,z) where x,y,z are integers with an even sum.
%C The L_infinity norm of a vector is the largest component in absolute value.
%C 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
%D J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.
%H R. J. Mathar, <a href="http://arxiv.org/abs/1002.3844">Point counts of D_k and some A_k and E_k integer lattices inside hypercubes</a> arXiv:1002.3844 [math.GT], 2010.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html">Home page for this lattice</a>
%H <a href="/index/Fa#fcc">Index entries for sequences related to f.c.c. lattice</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -2, 1).
%F From _R. J. Mathar_, Feb 03 2010: (Start)
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>4.
%F a(n) = 1 + (-1)^n + 12*n^2, n>0.
%F G.f.: 1 - 2*x*(6 + 13*x + 4*x^2 + x^3)/((1+x)*(x-1)^3). (End)
%e a(0) = 1: 000
%e a(1) = 12: +-1 +-1 0, where the 0 can be in any of the three coordinates
%e a(2) = 50: +-2 0 0 (6), +-2 +-1 +-1 (24), +-2 +-2 0 (12), +-2 +-2 +-2 (8).
%p 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
%t a[0] = 1; a[n_] := 1 + (-1)^n + 12*n^2;
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 16 2017, after _R. J. Mathar_ *)
%Y Cf. A117216, A022144, A010014, A175112 (D_5), A175114 (D_6).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Apr 15 2008
%E 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.
%E Extended by _R. J. Mathar_, Feb 03 2010
%E Removed the "conjectured" attribute from formulas - _R. J. Mathar_, Feb 27 2010
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