A008397 Coordination sequence for E_7 lattice.
1, 126, 2898, 25886, 133506, 490014, 1433810, 3573054, 7902594, 15942206, 29896146, 52834014, 88892930, 143501022, 223622226, 338022398, 497556738, 715478526, 1007769170, 1393489566
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 (7,-21,35,-35,21,-7,1).
Programs
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Magma
[1] cat [(2/5)*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n + 5): n in [1..30]]; // G. C. Greubel, May 29 2023
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Maple
a:= n-> `if`(n=0, 1, 148/5*n^6-12/5*n^5+52*n^4+12*n^3+212/5*n^2-48/5*n+2): seq(a(n), n=0..25);
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Mathematica
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,126,2898,25886,133506, 490014,1433810,3573054},20] (* Harvey P. Dale, Nov 12 2014 *) -
SageMath
[2*(74*n^6 -6*n^5 +130*n^4 +30*n^3 +106*n^2 -24*n +5)//5 - int(n==0) for n in range(31)] # G. C. Greubel, May 29 2023
Formula
a(n) = (2/5)*(74*n^6 - 6*n^5 + 130*n^4 + 30*n^3 + 106*n^2 - 24*n + 5) for n >= 1.
Bacher et al. give a g.f.
G.f.: (1 + 119*x + 2037*x^2 + 8211*x^3 + 8787*x^4 + 2037*x^5 + 119*x^6 + x^7)/(1-x)^7. - Colin Barker, Sep 26 2012