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%I #43 Dec 10 2023 16:06:07
%S 1,72,1062,6696,26316,77688,189810,405720,785304,1408104,2376126,
%T 3816648,5885028,8767512,12684042,17891064,24684336,33401736,44426070,
%U 58187880,75168252,95901624,120978594,151048728,186823368,229078440,278657262,336473352,403513236
%N Coordination sequence for E_6 lattice.
%D M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
%H T. D. Noe, <a href="/A008399/b008399.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_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
%F Bacher et al. give a g.f.
%F G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^6 = 1 + 18*x*(4+35*x+78*x^2+35*x^3+4*x^4)/(1-x)^6. - _Colin Barker_, Sep 26 2012
%F E.g.f.: 1 + (1/5)*x*(360 + 2295*x + 3105*x^2 + 1170*x^3 + 117*x^4 )*exp(x). - _G. C. Greubel_, May 29 2023
%p 1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);
%t LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1062,6696,26316,77688, 189810},30] (* _Harvey P. Dale_, Oct 24 2022 *)
%o (Magma) [1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // _G. C. Greubel_, May 29 2023
%o (SageMath) [9*n*(13*n^2+7)*(n^2+1)//5 +int(n==0) for n in range(41)] # _G. C. Greubel_, May 29 2023
%Y Cf. A008397, A008340, A019557, A019558.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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