login
Coordination sequence for {E_6}* lattice.
2

%I #33 Dec 10 2023 16:08:39

%S 1,54,828,5202,20376,60030,146484,312858,605232,1084806,1830060,

%T 2938914,4530888,6749262,9763236,13770090,18997344,25704918,34187292,

%U 44775666,57840120,73791774,93084948

%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="/A008401/b008401.txt">Table of n, a(n) for n = 0..1000</a>

%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) = 6*n*(3*n^4 + 5*n^2 + 1), n > 0.

%F G.f.: (1+48*x+519*x^2+1024*x^3+519*x^4+48*x^5+x^6)/(1-x)^6.

%F E.g.f.: 1 + 6*exp(x)*x*(9 + 60*x + 80*x^2 + 30*x^3 + 3*x^4). - _Stefano Spezia_, Apr 15 2022

%t Join[{1},Table[18 n^5+30 n^3+6 n,{n,30}]] (* _Harvey P. Dale_, May 16 2012 *)

%o (Magma) [1] cat [6*n*(3*n^4+5*n^2+1): n in [1..40]]; // _G. C. Greubel_, May 30 2023

%o (SageMath) [6*n*(3*n^4+5*n^2+1) +int(n==0) for n in range(41)] # _G. C. Greubel_, May 30 2023

%Y Cf. A008402 (partial sums).

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_