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%I #34 Dec 10 2023 16:12:17
%S 1,12,60,180,408,780,1332,2100,3120,4428,6060,8052,10440,13260,16548,
%T 20340,24672,29580,35100,41268,48120,55692,64020,73140,83088,93900,
%U 105612,118260,131880,146508,162180,178932,196800,215820,236028,257460,280152,304140,329460,356148,384240
%N Coordination sequence for 4-dimensional primitive di-isohexagonal orthogonal lattice.
%C Coordination sequence for 4-dimensional cyclotomic lattice Z[zeta_12].
%D M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
%H Vincenzo Librandi, <a href="/A008530/b008530.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Beck and S. Hosten, <a href="http://arxiv.org/abs/math/0508136">Cyclotomic polytopes and growth series of cyclotomic lattices</a>, arXiv:math/0508136 [math.CO], 2005-2006.
%H M. O'Keeffe, <a href="http://dx.doi.org/10.1524/zkri.1995.210.12.905">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908.
%H M. O'Keeffe, <a href="/A008527/a008527.pdf">Coordination sequences for lattices</a>, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: (1+4*x+x^2)^2/(1-x)^4. - _Colin Barker_, Apr 14 2012
%F 3*a(n) = (2*n+1)^3 + (2*n-1)^3 + (n+1)^3 + (n-1)^3 for n>0. - _Bruno Berselli_, Jan 31 2013
%F E.g.f.: 1 + x*(12 + 18*x + 6*x^2)*exp(x). - _G. C. Greubel_, Nov 10 2019
%e 3*a(5) = 2340 = (2*5+1)^3 + (2*5-1)^3 + (5+1)^3 + (5-1)^3. - _Bruno Berselli_, Jan 31 2013
%p 1, seq( 6*k^3+6*k, k=1..45);
%t CoefficientList[Series[(1+4*x+x^2)^2/(1-x)^4,{x,0,45}],x] (* _Vincenzo Librandi_, Apr 16 2012 *)
%t LinearRecurrence[{4,-6,4,-1}, {1,12,60,180,408}, 45] (* _G. C. Greubel_, Nov 10 2019 *)
%o (Magma) [1]cat[6*n^3+6*n: n in [1..45]]; // _Vincenzo Librandi_, Apr 16 2012
%o (PARI) vector(46, n, if(n==1,1, 6*(n-1)*(1+(n-1)^2)) ) \\ _G. C. Greubel_, Nov 10 2019
%o (Sage) [1]+[6*n*(1+n^2) for n in (1..45)] # _G. C. Greubel_, Nov 10 2019
%o (GAP) Concatenation([1], List([1..45], n-> 6*n*(1+n^2) )); # _G. C. Greubel_, Nov 10 2019
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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