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%I #35 Dec 10 2023 16:07:42
%S 1,73,1135,7831,34147,111835,301645,707365,1492669,2900773,5276899,
%T 9093547,14978575,23746087,36430129,54321193,79005529,112407265,
%U 156833335,215021215,290189467,386091091
%N Crystal ball 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="/A008400/b008400.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/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F a(n) = 1 + (3/20)*n*(n+1)*(26*n^4 + 52*n^3 + 73*n^2 + 47*n + 42).
%F G.f.: (1+66*x+645*x^2+1384*x^3+645*x^4+66*x^5+x^6)/(1-x)^7. - _Colin Barker_, Mar 16 2012
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - _Harvey P. Dale_, Feb 23 2015
%F E.g.f.: (1/20)*(20 + 1440*x + 9900*x^2 + 15480*x^3 + 7785*x^4 + 1404*x^5 + 78*x^6)*exp(x). - _G. C. Greubel_, May 30 2023
%p seq(39/10*n^6+117/10*n^5+75/4*n^4+18*n^3+267/20*n^2+63/10*n+1, n=0..35);
%t Table[39/10 n^6+117/10 n^5+75/4 n^4+18n^3+267/20 n^2+63/10 n+1, {n,0, 30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,73,1135, 7831,34147,111835,301645},30] (* _Harvey P. Dale_, Feb 23 2015 *)
%o (PARI) a(n)=(78*n^6 + 234*n^5 + 375*n^4 + 360*n^3 + 267*n^2 + 126*n + 20)/20 \\ _Charles R Greathouse IV_, Feb 10 2017
%o (Magma) [1 +3*n*(n+1)*(26*n^4 +52*n^3 +73*n^2 +47*n +42)/20: n in [0..40]]; // _G. C. Greubel_, May 30 2023
%o (SageMath) [1 +3*n*(n+1)*(26*n^4 +52*n^3 +73*n^2 +47*n +42)//20 for n in range(41)] # _G. C. Greubel_, May 30 2023
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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