A008391 Coordination sequence for A_8 lattice.
1, 72, 1332, 11832, 66222, 271224, 889716, 2476296, 6077196, 13507416, 27717948, 53265960, 96900810, 168278760, 280819260, 452715672, 708113304, 1078467624, 1604095524, 2335932504, 3337508646, 4687156248, 6480461988, 8832976488, 11883194148, 15795816120
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- M. Baake and U. Grimm, Coordination sequences for root lattices and related graphs, arXiv:cond-mat/9706122, 1997; Zeit. f. Kristallographie, 212 (1997), 253-256.
- R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
- J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Crossrefs
Row 8 of A103881.
Programs
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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
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Maple
1, seq(n*(715*n^6 + 6006*n^4 +10395*n^2 +3044)/280, n=1..40);
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Mathematica
Join[{1},Table[143/56n^7+429/20n^5+297/8n^3+761/70n,{n,30}]] (* or *) 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 *) -
Maxima
a[0]:1$ a[1]:72$ a[2]:1332$ a[3]:11832$ a[4]:66222$ a[5]:271224$ a[6]:889716$ a[7]:2476296$ a[8]:6077196$ 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]; makelist(a[n],n,0,30); /* Martin Ettl, Oct 26 2012 */
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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
Formula
a(n) = n*(715*n^6 + 6006*n^4 + 10395*n^2 + 3044)/280, a(0) = 1.
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
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