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Coordination sequence for A_8 lattice.
2

%I #46 May 27 2023 03:15:32

%S 1,72,1332,11832,66222,271224,889716,2476296,6077196,13507416,

%T 27717948,53265960,96900810,168278760,280819260,452715672,708113304,

%U 1078467624,1604095524,2335932504,3337508646,4687156248,6480461988,8832976488,11883194148,15795816120

%N Coordination sequence for A_8 lattice.

%H T. D. Noe, <a href="/A008391/b008391.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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

%F a(n) = n*(715*n^6 + 6006*n^4 + 10395*n^2 + 3044)/280, a(0) = 1.

%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - _Harvey P. Dale_, Mar 04 2012

%F G.f.: (1 + 64*x + 784*x^2 + 3136*x^3 + 4900*x^4 + 3136*x^5 + 784*x^6 + 64*x^7 + x^8)/(1-x)^8. - _Colin Barker_, Sep 26 2012

%p 1, seq(n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280, n=1..40);

%t Join[{1},Table[143/56n^7+429/20n^5+297/8n^3+761/70n,{n,30}]] (* or *)

%t Join[{1},LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{72,1332,11832, 66222,271224,889716,2476296,6077196},30]] (* _Harvey P. Dale_, Mar 04 2012 *)

%o (Maxima)

%o a[0]:1$

%o a[1]:72$

%o a[2]:1332$

%o a[3]:11832$

%o a[4]:66222$

%o a[5]:271224$

%o a[6]:889716$

%o a[7]:2476296$

%o a[8]:6077196$

%o a[n]:=8*a[n-1]-28*a[n-2]+ 56*a[n-3]- 70*a[n-4]+56*a[n-5]-28*a[n-6]+8*a[n-7]-a[n-8];

%o makelist(a[n],n,0,30); /* _Martin Ettl_, Oct 26 2012 */

%o (Magma) [1] cat [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280: n in [1..40]]; // _G. C. Greubel_, May 26 2023

%o (SageMath) [n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)//280 +int(n==0) for n in range(41)] # _G. C. Greubel_, May 26 2023

%Y Row 8 of A103881.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_ and _J. H. Conway_