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A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.

%I A117216 #30 Aug 19 2025 09:34:30
%S A117216 1,40,272,888,2080,4040,6960,11032,16448,23400,32080,42680,55392,
%T A117216 70408,87920,108120,131200,157352,186768,219640,256160,296520,340912,
%U A117216 389528,442560,500200,562640,630072,702688,780680,864240,953560,1048832,1150248,1258000,1372280
%N A117216 Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.
%C A117216 This lattice consists of all points (w,x,y,z) where w,x,y,z are integers with an even sum.
%C A117216 The L_infinity norm of a vector is the largest component in absolute value.
%C A117216 Equals binomial transform of [1, 39, 193, 191, 1, -1, 1, -1, 1, ...]. - _Gary W. Adamson_, Feb 05 2010
%D A117216 J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Chap. 4.
%H A117216 Vincenzo Librandi, <a href="/A117216/b117216.txt">Table of n, a(n) for n = 0..1000</a>
%H A117216 Gabriele 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 A117216 <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>.
%H A117216 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A117216 From _R. J. Mathar_, Feb 03 2010, Feb 13 2010: (Start)
%F A117216 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>4;
%F A117216 a(n) = 8*n*(1+4*n^2) = 2*A144965(n), n>0 (bisection of A035878 and A105374). (End)
%F A117216 G.f.: (1 + 36*x + 118*x^2 + 36*x^3 + x^4)/(1-x)^4. - _Colin Barker_, May 24 2012
%F A117216 E.g.f.: 1 + 8*x*(1 + 2*x)*(5 + 2*x)*exp(x). - _Elmo R. Oliveira_, Aug 18 2025
%t A117216 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 A117216 (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 A117216 Cf. A035878, A105374, A110907, A144965, A175110.
%K A117216 nonn,easy,changed
%O A117216 0,2
%A A117216 _N. J. A. Sloane_, Apr 15 2008
%E A117216 a(2) corrected and sequence extended by _R. J. Mathar_, Feb 03 2010, Feb 13 2010