A008399 Coordination sequence for E_6 lattice.
1, 72, 1062, 6696, 26316, 77688, 189810, 405720, 785304, 1408104, 2376126, 3816648, 5885028, 8767512, 12684042, 17891064, 24684336, 33401736, 44426070, 58187880, 75168252, 95901624, 120978594, 151048728, 186823368, 229078440, 278657262, 336473352, 403513236
Offset: 0
References
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
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).
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908.
- M. O'Keeffe, Coordination sequences for lattices, Zeit. f. Krist., 210 (1995), 905-908. [Annotated scanned copy]
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
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Magma
[1] cat [9*n*(13*n^2+7)*(n^2+1)/5: n in [1..40]]; // G. C. Greubel, May 29 2023
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Maple
1, seq(117/5*n^5+36*n^3+63/5*n, n=1..30);
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Mathematica
LinearRecurrence[{6,-15,20,-15,6,-1},{1,72,1062,6696,26316,77688, 189810},30] (* Harvey P. Dale, Oct 24 2022 *) -
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
Formula
a(n) = 9*n*(13*n^2+7)*(n^2+1)/5 for n >= 1.
Bacher et al. give a g.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
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