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%I #26 Sep 08 2022 08:45:24
%S 1,40,272,888,2080,4040,6960,11032,16448,23400,32080,42680,55392,
%T 70408,87920,108120,131200,157352,186768,219640,256160,296520,340912,
%U 389528,442560,500200,562640,630072,702688,780680,864240,953560,1048832,1150248
%N Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.
%C This lattice consists of all points (w,x,y,z) where w,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 Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Feb 05 2010
%D J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.
%H Vincenzo Librandi, <a href="/A117216/b117216.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html">Home page for this lattice</a>
%H <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F From _R. J. Mathar_, Feb 03 2010, Feb 13 2010: (Start)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
%F a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
%F G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - _Colin Barker_, May 24 2012
%t CoefficientList[Series[(1+36*x+118*x^2+36*x^3+x^4)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 27 2012 *)
%o (Magma) I:=[1, 40, 272, 888, 2080]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 27 2012
%Y Cf. A110907, A175110.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Apr 15 2008
%E a(2) corrected and sequence extended by _R. J. Mathar_, Feb 03 2010, Feb 13 2010
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