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%I #43 Dec 10 2023 16:05:19
%S 1,20,110,340,780,1500,2570,4060,6040,8580,11750,15620,20260,25740,
%T 32130,39500,47920,57460,68190,80180,93500,108220,124410,142140,
%U 161480,182500,205270,229860,256340,284780,315250,347820,382560,419540,458830,500500,544620
%N Coordination sequence for A_4 lattice.
%D M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
%H T. D. Noe, <a href="/A008383/b008383.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Baake and U. Grimm, <a href="https://arxiv.org/abs/cond-mat/9706122">Coordination sequences for root lattices and related graphs</a>, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="https://doi.org/10.1016/S0764-4442(97)83542-2">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%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 a(n) = 5*n*(7*n^2 + 5)/3, a(0) = 1.
%F G.f.: (1+16*x+36*x^2+16*x^3+x^4)/(1-x)^4 = 1+10*x*(2+3*x+2*x^2)/(x-1)^4. - _Colin Barker_, Apr 13 2012
%F E.g.f.: (1/3)*(3 + 5*x*(12 + 21*x + 7*x^2)*exp(x)). - _G. C. Greubel_, May 25 2023
%p a:= n-> `if`(n=0, 1, 35/3*n^3+25/3*n): seq (a(n), n=0..50);
%t CoefficientList[Series[(1+16x+36x^2+16x^3+x^4)/(1-x)^4,{x,0,40}],x] (* _Harvey P. Dale_, Dec 01 2013 *)
%t Join[{1}, LinearRecurrence[{4, -6, 4, -1}, {20, 110, 340, 780}, 40]] (* _Jean-François Alcover_, Jan 07 2019 *)
%o (Magma) [n eq 0 select 1 else 5*n*(7*n^2+5)/3: n in [0..45]]; // _G. C. Greubel_, May 25 2023
%o (SageMath) [5*n*(7*n^2+5)/3+int(n==0) for n in range(46)] # _G. C. Greubel_, May 25 2023
%Y Cf. A005901, A008384, A008385.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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