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%I #35 May 30 2023 07:46:37
%S 1,127,3025,28911,162417,652431,2086241,5659295,13561889,29504095,
%T 59400241,112234255,201127185,344628207,568250433,906272831,
%U 1403829569,2119308095,3127077265,4520566831
%N Crystal ball sequence for E_7 lattice.
%H T. D. Noe, <a href="/A008398/b008398.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 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>
%F a(n) = 1 + (2/105)*n*(222*n^6 + 756*n^5 + 1260*n^4 + 1470*n^3 + 1708*n^2 + 1029*n + 170).
%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)^8. - _Colin Barker_, Mar 16 2012
%F 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_, Jul 18 2012
%p seq(148/35*n^7+72/5*n^6+24*n^5+28*n^4+488/15*n^3+98/5*n^2+68/21*n+1, n=0..30);
%t Table[148/35n^7+72/5n^6+24n^5+28n^4+488/15n^3+98/5n^2+68/21n+1,{n,0,20}] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,127,3025,28911,162417,652431,2086241,5659295},20] (* _Harvey P. Dale_, Jul 18 2012 *)
%o (PARI) a(n)=(444*n^7 + 1512*n^6 + 2520*n^5 + 2940*n^4 + 3416*n^3 + 2058*n^2 + 340*n + 105)/105 \\ _Charles R Greathouse IV_, Feb 10 2017
%o (Magma) [1 +(2/105)*n*(222*n^6 +756*n^5 +1260*n^4 +1470*n^3 +1708*n^2 +1029*n +170): n in [0..30]]; // _G. C. Greubel_, May 29 2023
%o (SageMath)
%o def A008398(n): return 1 +2*n*(222*n^6 +756*n^5 +1260*n^4 +1470*n^3 +1708*n^2 +1029*n +170)//105
%o [A008398(n) for n in range(31)] # _G. C. Greubel_, May 29 2023
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_ and _J. H. Conway_
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